•uJilMBjja.j, 


D.  VAN  NOSTEAND,  PUBLISHEB, 

16    MURRAY    STREET    AND    V7    WARRKN    STREBT, 

NEW  YORK. 


OP  THE 

TJ&I7ERSIT7 


A  'TEXT   BOOK 


OF 


DRAWING; 


FOR  THE  USE  OF 

MECHANICS   AND   SCHOOLS, 

IV   WHICH   TITR    DEFINITIONS   AND    RULES    OF  GEOMETRY  ARE   FAMILIARLY    EXPLAINED,  TI13 

PRACTICAL    PROBLEMS  ARK  ARRANGED    FROM    THE    MOST  SIMPLE  TO  THE    MORE 

COMPLEX,  AND  IN  THEIR    DESCRIPTION  TECHNICALITIES    ARE 

AVOIDED    AS   MUCH    AS    POSSIBLE; 

WITH    ILLUSTRATIONS    FOR    DRAWING- 

PLANS,  SECTIONS  AND  ELEVATIONS  OF  BUILDINGS  AND  MACHINERY  . 

AN  INTRODUCTION  TO  ISO-METRICAL  DRAWING  : 

A    COURSE    OF 

LINEAR   PERSPECTIVE   AND   SHADOWS: 

AN    ESSAY    ON 

THE    THEORY    OF  ' 


YV  *   •    »»s  -  » 

ITS   APPLICATION    TO   ARCHITECTURAL   AND    MECHANlCXlV  I/RA  vVVNGS 

THE    WHOLE    ILLUSTRATED    WITH 

STIEIEJILi 


BY  WM.  MTNIF1E,  ARCHITECT. 


NINTH    THOUSAND,    REVISED    BY    THE    AUTHOR, 


NEW    YORK: 
D.    VAN    NOSTEAND,    PUBLISHES, 

23  MURRAY  STREET  AND  27  WARREN  STREET. 


09  THB  " 


*>       v^ 

y     V) 


Entered,  according  to  the  Act  of  Congress,  in  the  year  1867 

BY    WM.    M  1  N  I  F  I  E, 
In  the  Clerk's  Office  of  the  District  Court  of  Maryland 


PREFACE. 


HAVING  been  for  several  years  engaged  in  teaching  Architectural  and  Me- 
chanical drawing,  both  in  the  High  School  of  Baltimore  and  to  private  classes, 
I  have  endeavored  without  success,  to  procure  a  book  that  I  could  introduce 
as  a  text  book,  works  on  Geometry  generally  contain  too  much  theory  for 
the  purpose,  with  an  insufficient  amount  of  practical  problems ;  and  books  on 
Architecture  and  Machinery  are  mostly  too  voluminous  and  costly,  contain- 
ing much  that  is  entirely  unnecessary  for  the  purpose.  Under  these  circum- 
stances, I  collected  most  of  the  useful  practical  problems  in  geometry  from  a 
variety  of  sources,  simplified  them  and  drew  them  on  cards  for  the  use  of  the 
classes,  arranging  them  from  the  most  easy  to  the  more  difficult,  thus  leading 
the  students  gradually  forward ;  this  was  followed  by  the  drawing  of  plans, 
sections,  elevations  and  details  of  Buildings  and  Machinery,  then  followed 
Isometrical  drawing,  and  the  course  was  closed  by  the  study  of  Linear  per- 
spective and  shadows;  the  whole  being  illustrated  by  a  series  of  short  lectures 
to  the  private  classes. 

I  have  been  so  well  pleased  with  the  results  of  this  method  of  instruction, 
that  I  have  endeavored  to  adopt  its  general  features  in  the  arrangement  of 
the  following  work.  The  problems  in  constructive  geometry  have  been  selected 
with  a  view  to  their  practical  application  in  the  every-day  business  of  the 
Engineer,  Architect  and  Artizan,  while  at  the  same  time  they  afford  a  good 
series  of  lessons  to  facilitate  the  knowledge  and  use  of  the  instruments  requir- 
ed in  mechanical  drawing. 

The  definitions  and  explanations  have  been  given  in  as  plain  and  simple 
language  as  the  subject  will  admit  of;  many  persons  will  no  doubt  think 
them  too  simple.  Had  the  book  been  intended  for  the  use  of  persons  versed 
in  geometry,  very  many  of  the  explanations  might  have  been  dispensed  with, 
but  it  is  intended  chiefly  to  be  used  as  a  first  book  in  geometrical  drawing ,  by 
persons  who  have  not  had  the  benefit  of  a  mathematical  education,  and  who 
in  a  majority  of  cases,  have  not  the  time  or  inclination  to  study  any  com- 
plex matter,  or  what  is  the  same  thing,  that  which  may  appear  so  to  them. 
Arid  if  used  in  schoolsjits  detailed  explanations,  we  believe,  will  save  time  to 
the  teacher,  by  permitting  the  scholar  to  obtain  for  himself  much  information 
that  he  would  otherwise  require  to  have  explained  to  him. 

But  it  is  also  intended  to  be  used  for  self -instruction ,  without  the  aid  of  a 
teacher,  to  whom  the  student  might  refer  for  explanation  of  any  difficulty; 
under  these  circumstances  I  do  not  believe  an  explanation  can  be  couched  in 
too  simple  language.  With  a  view  of  adapting  the  book  to  this  class  of  stu- 
dents, the  illustrations  of  each  branch  treated  of,  have  been  made  progressive, 
commencing  with  the  plainest  diagrams;  and  even  in  the  more  advanced,  the 
object  has  been  to  instil  principles  rather  than  to  produce  effect,  as  those  once 


IV 

obtained,  the  student  can  either  design  for  himself  or  copy  from  any  subject 
at  hand.  It  is  hoped  that  this  arrangement  will  induce  many  to  study  draw- 
ing who  would  not  otherwise  have  attempted  it,  and  thereby  render  them- 
selves much  more  capable  of  conducting  any  business,  for  it  has  been  truly 
said  by  an  eminent  writer  on  Architecture,  "  that  one  workman  is  superior  to 
another  (other  circumstances  being  the  same)  directly  in  proportion  to  his 
knowledge  of  drawing,  and  those  who  are  ignorant  of  it  must  in  many  re- 
spects be  subservient  to  others  who  have  obtained  that  knowledge." 

The  size  of  the  work  has  imperceptibly  increased  far  beyond  my  original 
design,  which  was  to  get  it  up  in  a  cheap  form  with  illustrations  on  wood, 
and  to  contain  about  two-thirds  of  the  number  in  the  present  volume,  but  on 
examining  some  specimens  of  mathematical  diagrams  executed  on  wood,  I 
was  dissatisfied  with  their  want  of  neatness,  particularly  as  but  few  students 
aim  to  excel  their  copy.  On  determining  to  use  steel  illustrations  I  deemed 
it  advisable  to  extend  its  scope  until  it  has  attained  its  present  bulk,  and  even 
now  I  feel  more  disposed  to  increase  than  to  curtail  it,  as  it  contains  but  few 
examples  either  in  Architecture  or  Machinery.  I  trust,  however,  that  the 
objector  to  its  size  will  find  it  to  contain  but  little  that  is  absolutely  useless  to 
a  student. 

In  conclusion,  I  must  warn  my  readers  against  an  idea  that  I  am  sorry  to 
find  too  prevalent,  viz  :  that  drawing  requires  but  little  time  or  study  for  its 
attainment,  that  it  may  be  imbibed  involuntarily  as  one  would  fragrance  in  a 
flower  garden,  with  little  or  no  exertion  on  the  part  of  the  recipient,  not  that 
the  idea  is  expressed  in  so  many  words,  but  it  is  frequently  manifested  by  their 
dissatisfaction  at  not  being  able  to  make  a  drawing  in  a  few  lessons  as  well 
is  their  teacher,  even  before  they  have  had  sufficient  practice  to  have  obtained 
a  free  use  of  the  instruments.  I  have  known  many  give  up  the  study  in  con- 
equence,  who  at  the  same  time  if  they  should  be  apprenticed  to  a  carpenter, 
yould  be  satisfied  if  they  could  use  the  jack  plane  with  facility  after  several 
weeks  practice,  or  be  able  to  make  a  sash  at  the  end  of  some  years. 

Now  this  idea  is  fallacious,  and  calculated  to  do  much  injury;  proficiency 
n  no  art  can  be  obtained  without  attentive  study  and  industrious  persever- 
ance. Drawing  is  certainly  not  an  exception :  but  the  difficulties  will  soon 
anish  if  you  commence  with  a  determination  to  succeed ;  let  your  motto  be 
PERSEVERE,  never  say  "it  is  too  difficult;"  you  will  not  find  it  so  difficult 
as  you  imagine  if  you  will  only  give  it  proper  attention ;  and  if  my  labors 
lave  helped  to  smooth  those  difficulties  it  will  be  to  me  a  source  of  much 
gratification. 

WM.  MINIFIE. 
BALTIMORE,  1st  March,  1849. 


PREFACE 

TO    THE    REVISED    EDITION 


IN  issuing  this  seventh  edition  of  THE  GEOMETRICAL  DRAWING  BOOK,  the 
author  desires  to  return  his  grateful  thanks  to  the  public,  for  the  favor  with 
which  the  previous  editions  have  been  received :  more  especially  for  the  favor- 
able notices  elicited  from  the  press,  both  in  the  United  States  and  Great 
Britain,  particularly  from  that  portion  devoted  to  Fine  Art,  Architecture,  En- 
gineering and  Mechanics,  as  in  all  those  pursuits  a  knowledge  of  drawing  is 
indispensable  to  success,  and  the  conductors  of  its  literature  may  be  fairly  con- 
sidered as  the  most  competent  to  decide  on  the  merit  of  a  treatise  on  this  sub- 
ject ;  their  approval  has,  therefore,  afforded  me  the  more  satisfaction. 

Many  of  the  schools  and  colleges  of  the  Union  have  adopted  the  work  as  a 
Text  Book.  It  has  also  been  recommended  by  the  Department  of  Art  of  the 
British  Government  to  the  National  and  other  Public  Schools  and  Institutions 
throughout  the  kingdom. 

The  present  edition  has  been  carefully  examined  and  the  few  typographical 
errors  corrected.  The  Essay  on  the  Theory  of  Color  and  its  application  to 
Architectural  and  Mechanical  Drawings,  which  was  issued  as  an  appendix  to 
the  fourth  and  following  editions,  has  now  been  revised  and  arranged  in  the 
body  of  the  work,  and  a  full  Index  to  the  Essay  added,  together  with  a  few 
other  matters  of  interest,  which  it  is  hoped  will  be  found  to  add  to  the  usefulness 
of  the  book,  and  enable  the  student  much  more  readily  to  refer  to  the  informa- 
tion required. 

BALTIMORE,  J/ay,  1867. 


ILLUSTRATIONS. 


PLATE 


Definitions  of  lines  and  angles,          .......         i. 

Definitions  of  plane  rectilinear  superficies, ii. 

Definitions  of  the  circle, iii. 

To  erect  or  let  fall  a  perpendicular, iv. 

Construction  and  division  of  angles, v. 

Construction  of  polygons, vi. 

Problems  relating  to  the  circle,          .         .         ,  .         .         .      vii. 

Parallel  ruler,  and  its  application, viii. 

Scale  of  chords  and  plane  scales, ix. 

Protractor,  its  construction  and  application,  ....  x. 

Flat  segments  of  circles  and  parabolas, xi. 

Oval  figures  composed  of  arcs  of  circles,        .         .  .         .  xii. 

Cycloid  and  Epicycloid, xiii. 

Cube,  its  sections  and  surface,     .......          xiv. 

Prisms,  square  pyramid  and  their  coverings,       .         .         .         .         .      xv. 

Pyramid,  Cylinder,  Cone  and  their  surfaces,  ....          xvi. 

Sphere  and  covering  and  coverings  of  the  regular  Polyhedrons,  .         .    xvii. 
Cylinder  and  its  sections,    ........        xviii. 

Cone  and  its  sections, xix. 

Ellipsis  and  Hyperbola,       .         .         ...         .         .         .  xx. 

Parabola  and  its  application  to  Gothic  arches, xxi. 

To  find  the  section  of  the  segment  of  a  cylinder  through  three  given 

points,    ..........         xxii. 

Coverings  of  hemispherical  domes, xxiii. 

Joints  in  circular  and  elliptic  arches,     ......        xxiv. 

Joints  in  Gothic  arches,  .........    xxv. 

Design  for  a  Cottage — ground  plan  and  elevation,       .         .         .  xxvi. 

Design  for  a  Cottage — chamber  plan  and  section,   ....      xxvii. 

Details  of  a  Cottage — -joists,  roof  and  cornice,         ....    xxviii. 

Details  of  a  Cottage — parlor  windows  and  plinth,        .  .  xxix. 

Octagonal  plan  and  elevation, xxx. 

Circular  plan  and  elevation,      .......  xxxi. 

Roman  mouldings,      ....  ....       xxxii. 

Grecian  mouldings,          .         .         .         .  .         .         .          xxxiii. 

Plan,  section  and  elevation  of  a  wheel  and  pinion,          .         .         .      xxxiv. 
To  proportion  the  teeth  of  wheels,  ......  xxxv. 

Cylinder  of  a  locomotive,  plan  and  section,  ....      xxxvi. 

Cylinder  of  a  locomotive,  transverse  section  and  end  view,          .          xxxvii 


VI 


Isometrical  cube,  its  construction, xxxviii. 

Isometrical  figures,  triangle  and  square,  ....  xxxix. 
Isometrical  figures  pierced  and  chamfered,  .  .  xl. 

Isometrical  circle,  method  of  describing  and  dividing  it,  .  .  .  xli. 
Perspective— Visual  angle,  section  of  the  eye,  &c xlii. 

"          Foreshortening  and  definitions  of  lines,       .         .         .         xliii. 

"  Squares,  half  distance,  and  plan  of  a  room,   .         .         .    xliv. 

"          Tessellated  pavements, xlv. 

"          Square  viewed  diagonally.     Circle xlvi. 

"          Line  of  elevation,  pillars  and  pyramids,      .         .         .        xlvii. 

"          Arches  parallel  to  the  plane  of  the  picture,      .         .         .  xlviii. 

"          Arches  on  a  vanishing  plane,. xlix. 

"          Application  of  the  circle,  .  .  1. 

«          Perspective  plane  and  vanishing  points,  .         .  li. 

"  Cube  viewed  accidentally,     ......       Hi. 

"  Cottage  viewed  accidentally, Hii. 

"  Frontispiece.  Street  parallel  to  the  middle  visual  ray,  liv. 

Shadows,  rectangular  and  circular, lv. 

Shadows  of  steps  and  cylinder,  ......  Ivi. 


DEFINITIONS  Oh'  LINES  AN11  ANdf.KS. 


0*  THB 

TJKIVBRSIT7 


PRACTICAL   GEOMETRY, 

PLATE  I. 

DEFINITIONS   OF  LINES  AND  ANGLES. 


1.  A  POINT  is  said  to  have  position  without  magnitude;  and  it  is 
therefore  generally  represented  to  the  eye  by  a  small  dot,  as  at  A. 

2.  A  LINE  is  considered  as  length  without  breadth  or  thickness, 
it  is  in  fact  a  succession  of  points;  its  extremities  therefore,  are 
points.     Lines  are  of  three  kinds  ;   right  lines,  curved  lines,  and 
mixed  lines. 

3.  A  RIGHT  LINE,  or  as  it  is  more  commonly  called,  a  straight 
line,  is  the  shortest  that  can  be  drawn  between  two  given  points 
as  B. 

4.  A  CURVE  or  CURVED  LINE  is  that  which  does  not  lie  evenly 
between  its  terminating  points,  and  of  which  no  portion,  how- 
ever small,  is  straight  ;   it  is  therefore  longer  than  a  straight  line 
connecting  the  same  points.     Curved  lines  are  either  regular  or 
irregular. 

5.  A  REGULAR  CURVED  LINE,  as    C;  is  a  portion  of  the  circum- 
ference of  a  circle,  the  degree  of  curvature   being   the   same 
throughout    its    entire    length.     An   irregular  curved    line  has 
not  the  same  degree  of  curvature  throughout,  but  varies  at  dif- 
ferent points. 

6.  A  WAVED  LINE  may  be  either  regular  or  irregular  ;  it  is  com- 
posed  of  curves  bent  in  contrary  directions,  j  ^  Jis  &  regular 
waved  line,  the  inflections  on  either  side  of  .tbe  ,dott$<J,)ine  >  bring 

' 


equal;  a  waved  line  is  also  called  a  line 
contrary  flexure,  and  a  serpentine  line. 

7.  MIXED  LINES  are  composed  of  straight  and  curved  lines,  as  D. 

8.  PARALLEL  LINES  are  those  which  have  no  inclination  to  t:ich 
other,  as  F,  being  every  where  equidistant;  consequently  they 
could  never  meet,  though  produced  to  infinity. 


8  PLATE    I. 

If  the  parallel  lines  G  were  produced,  they  would  form  two 
concentric  circles,  viz:  circles  which  have  a  common  centre, 
whose  boundaries  are  every  where  parallel  and  equidistant. 

9.  INCLINED  LINES,  as  H  and   /,  if  produced,  would  meet  in  a 
point  as  at   K,  forming  an  angle  of  which  the  point   K  is  called 
the  vertex  or  angular  point,  and  the  lines  H  and  /  the  legs  or 
sides  of  the  angle    K;    the  point  of  meeting  is  also  called  the 
summit  of  an  angle. 

10.  PERPENDICULAR  LINES. — Lines  are  perpendicular  to  each 
other  when  the  angles  on  either  side  of  the  point  of  junction  are 
equal;  thus  the  lines  JV.  0.  P  are  perpendicular  to  the  line  L 
M.     The  lines  N.  0.  P  are  called  also  vertical  lines  and  plumb 
lines,  because  they  are  parallel  with  any  line  to  which  a  plummet 
is  suspended ;  the  line  L.  JV/  is  a  horizontal  or  level  line ;  lines 
so  called  are  always  perpendicular  to  a  plumb  line. 

11.  VERTICAL  and  HORIZONTAL  LINES  are  always  perpendicular 
to  each  other,  but  perpendicular  lines  are  not  always  vertical  and 
horizontal;  they  may  be  at  any  inclination  to  the  horizon  pro- 
vided that  the  angles  on  either  side  of  the  point  of  intersection 
are  equal,  as  for  example  the  lines  X.   Y  and  Z. 

12.  ANGLES. — Two  right  lines  drawn  from  the  same  point,  di- 
verging from  each  other,  form  an  angle,  as  the  lines  S.    Q.  R. 
An  angle  is  commonly  designated  by  three  letters,  and  the  letter 
designating  the  point  of  divergence,  which  in  this  case  is   Q,  is 
always  placed  in  the  middle.     Angles  are  either  acute,  right  or 
obtuse.    If  the  legs  of  an  angle  are  perpendicular  to  each  other, 
they  form  a  right  angle  as  T.  Q.  R,  (mechanics'  squares,  if  true, 
are  always  right  angled;)  if  the  sides  are  nearer  together,  as  S. 
Q.  R,  they  form  an  acute  angle;  if  the  sides  are  wider  apart,  or 
diverge  from  each  other  more  than  a  right  angle,  they  form  an 
obtuse  angle,  as   V.  Q.  R. 

The  magnitude  of  an  angle  does  not  depend  on  the  length  of 
the  sides,  but  upon  their  divergence  from  each  other ;  an  angle  is 
rsaidr  to  be  greater  or  less  than  another  as  the  divergence  is  greater 
orrless;rth^srtherrbUuse  angle   V.  Q.  R  is  greater,  and  the  acute 
than  the  right  angle  T.  Q.  R. 


UNIVERSITY) 


DEFINITIONS.  PLANE  RECTILINEAR  SUPERFICIES. 


TRIANGLES   OR  TRIGONS. 


C  d 


QUADRILATERALS,    QUADRANGLES  OR  TETRAGONS. 


PARALLELOGRAMS. 


Male  .7 
DEFINITIONS  OF  THE  CIRCLE. 


SEMICIRCLE. 


SEGMENTS. 

c, 


I '         A ! 


5. 

QUADRANT. 


COMPLEMENT. 

E 


7 

SUPPLE  Ml-:.\-T. 


9. 

SINE. 


ro-; 

L 


9 


PLATE   II. 

PLANE    RECTILINEAR    SUPERFICIES, 


13.  A  SUPERFICIES  or  SURFACE  is  considered  as  an  extension  of 
length  and  breadth  without  thickness. 

14.  A  PLANE  SUPERFICIES  is  an  enclosed  flat  surface  that  will 
coincide  in  every  place  with  a  straight  line.    It  is  a  succession  of 
straight  lines,  or  to  be  more  explicit,  if  a  perfectly  straight  edged 
ruler  be  placed  on  a  plane  superficies  in  any  direction,  it  would 
touch  it  in  every  part  of  its  entire  length. 

15.  When  surfaces  are  bounded  by  right  lines,  they  are  said  to  be 
RECTILINEAR  or  RECTILINEAL.,   As  all  the  figures   on   plate 
second  agree  with  the  above  definitions,  they  are  PLANE  RECTI- 
LINEAR SUPERFICIES. 

16.  Figures  bounded  by  more  than  four  right  lines  are   called 
POLYGONS;  the  boundary  of  a  polygon  is  called  its  PERIMETER. 

17.  When  SURFACES  are  bounded  by  three  right  lines,  they  are 
called  TRIANGLES  or  TRIGONS. 

18.  AN  EQUILATERAL  TRIANGLE  has  all  its  sides  of  equal  length, 
and  all  its  angles  equal,  as  Jl. 

19.  AN  ISOSCELES  TRIANGLE  has  two  of  its  sides  and  two  of  its 
angles  equal,  as  B. 

20.  A  SCALENE  TRIANGLE  has  all  its  sides  and  angles  unequal, 
as   C. 

21.  AN  ACUTE  ANGLED  TRIANGLE  has  all  its  angles  acute,  as  Ji 
and  B. 

22.  A  RIGHT  ANGLED  TRIANGLE  has  one  right  angle;  the  side 
opposite  the  right  angle  is  called  the  hypothenuse;  the  other  sides 
are  called  respectively  the  base  and  perpendicular.     The  figures 
A.  B.  C,  are  each  divided  into  two  right  angled  triangles  by  the 
dotted  lines  running  across  them. 

23.  AN  OBTUSE  ANGLED  TRIANGLE  has  one  obtuse  angle,  as  C. 

24.  If  figures  A  and  B  were  cut  out  and  folded  on  the  dotted 
line  in  the  centre  of  each,  the  opposite  sides  would  exactly  coin- 
cide ;  they  are  therefore,  regular  triangles. 

25.  Any  of  the  sides  of  an  equilateral  or  scalene  triangle  may  be 
called  its  BASE,  but  in  the  Isosceles  triangle  the  side  which  is 


10  PLATE    II. 

unequal  is  so  called,  the  angle   opposite  the  base  is  called  the 
VERTEX. 

26.  THE  ALTITUDE  of  a  Triangle  is  the  length  of  a  perpendicular 
let  fall  from  its  vertex  to  its  base,  as  a.  A.  and  b.  B,  or  to  its  base 
extended,  as  d.  d,  figure   C. 

The  superficial  contents  of  a  Triangle  may  be  obtained  by  mul- 
tiplying the  altitude  by  one  half  the  base. 

27.  When  surfaces  are  bounded  by  four  right  lines,  they  are  called 
QUADRILATERALS,  QUADRANGLES  or  TETRAGONS;  either  of  the 
figures  D.  E.  F.  G.  H  and  K  may  be  called  by  either  of  those 
terms,  which  are  common  to  all  four-sided  right  lined    figures, 
although  each  has  its  own  proper  name. 

28.  When  a  Quadrilateral  has  its  opposite  sides  parallel  to  each 
other,  it  is  called  a  PARALLELOGRAM  ;  therefore  figures  D.  E.  F 
and   G  are  parallelograms. 

29.  When  all  the  angles  of  a  Tetragon  are  right  angles,  the  figure 
is  called  a  RECTANGLE,  as  figures  D  and  E. 

If  two  opposite  angles  of  a  Tetragon  are  right  angles,  the  others 
are  necessarily  right  too. 

30.  If  the  sides  of  a  Rectangle  are  all  of  equal  length,  the  figure 
is  called  a  SQUARE,  as  figure  D. 

31.  If  the  sides  of  a  Rectangle  are  not  all  of  equal  length,  two  of 
its  sides  being  longer  than  the  others,  as  figure  E,  it  is  called  an 
OBLONG. 

32.  When  the  sides  of  a  parallelogram  are  all  equal,  and  the  an- 
gles not  right  angles,  two  being  acute  and  the  others  obtuse,  as 
figure  F,  it  is  called  a  RHOMB,  or  RHOMBUS  ;  it  is  also  called  a 
DIAMOND,  and  sometimes  a  LOZENGE,  more  particularly  so  when 
the  figure  is  used  in  heraldry. 

33.  A  parallelogram  whose  angles  are  not  right  angles,  but  whose 
opposite  sides  are  equal,  as  figure   G,  is  called  a  RHOMBOID. 

34.  If  two  of  the  sides  of  a  Quadrilateral  are   parallel  to   each 
other  as  the  sides  H  and  0  in  fig.  H,  it  is  called  a  TRAPEZOID. 

35.  All  other  Quadrangles  are  called  TRAPEZIUMS,  the  term  being 
applied  to  all  Tetragons  that  have  no  two  sides  parallel,  as  K. 

NOTE.  The  terms  TRAPEZOID  and  TRAPEZIUM  are  applied  indiscrimin- 
ately by  some  writers  to  either  of  the  figures  H  and  K;  by  others,  fig.  H  is 
called  a  Trapezium  and  fig.  K  a  Trapezoid,  and  this  appears  to  be  the  more 
correct  method;  but  asTrapezoid  is  a  word  of  comparatively  modern  origin, 
I  have  used  it  as  it  is  most  generally  applied  by  modern  writers,  more  par- 
ticularly so  in  works  on  Architecture  and  Mechanics. 


PLATE    III.  11 

36.  A  DIAGONAL  is  a  line  running  across  a  Quadrangle,  connect- 
ing its  opposite  corners,  as  the  dotted  lines  in  figs.  D  and  F. 

NOTE. — I  have  often  seen  persons  who  have  not  studied  Geometry,  much 
confused, in  consequence  of  the  number  of  names  given  to  the  same  figure, 
as  for  example  fig.  D. 

1st.     It  is  a  plane  Figure — see  paragraph  14. 

2nd.  It  is  Rectilineal,  being  composed  of  right  lines. 

3rd.    It  is  a  Quadrilateral,  being  bounded  by  four  lines. 

4th.    It  is  a  Quadrangle,  having  four  angles. 

5th.    It  is  a  Tetragon,  having  four  sides. 

6th.    It  is  a  Parallelogram,  its  opposite  sides  being  parallel. 

7th.    It  is  a  Rectangle,  all  its  angles  being  right  angles. 

All  the  above  may  be  called  common  names,  because  they  are  applied  to  all 
figures  having  the  same  properties. 

8th.  It  is  a  Square,  which  is  its  proper  name,  distinguishing  it  from  all  other 
figures,  to  which  some  or  all  of  the  above  terms  may  be  applied. 

All  of  them  except  7  and  8,  may  also  be  applied  to  fig.  F,  w:th  the  same 
propriety  as  to  fig.  D  ;  besides  these,  fig.  F  has  four  proper  names  distin- 
guishing it  from  all  other  figures,  viz:  a  Rhomb,  Rhombus,  Diamond  and 
Lozenge. 

If  the  student  will  analyze  all  the  other  figures  in  the  same  manner,  he 
will  soon  become  perfectly  familiar  with  them,  and  each  term  will  convey  to 
his  mind  a  clear  definite  idea. 


PLATE   III. 

DEFINITIONS    OF     THE    CIRCLE. 


1st.  A  CIRCLE  is  a  plane  figure  bounded  by  one  curve  line, 
every  where  equidistant  from  its  centre,  as  fig.  1. 

2nd.  The  boundary  line  is  called  the  CIRCUMFERENCE  or  PE- 
RIPHERY, it  is  also  for  convenience  called  a  Circle. 

3rd.  The  CENTRE  of  a  circle  is  a  point  within  the  circumference, 
equally  distant  from  every  point  in  it,  as  C,  fig.  1. 

4th.  The  RADIUS  of  a  circle  is  a  line  drawn  from  the  centre  to 
any  point  in  the  circumference,  as  C.  Jl,  O.  B  or  C.  D,  fig.  1. 

The  plural  of  Radius  is  RADII.  All  radii  of  the  same  circle  are 
of  equal  length. 

5th.  The  DIAMETER  of  a  circle  is  any  right  line  drawn  through 
the  centre  to  opposite  points  of  the  circumference,  as  Ji.  B}  fig.  1. 


12  PLATE    III. 

*• 

The  length  of  the  diameter  is  equal  to  two  radii;  there  may 
be  an  infinite  number  of  diameters  in  the  same  circle,  but  they 
are  all  equal. 

6th.  A  SEMICIRCLE  is  the  half  of  a  circle,  as  fig.  2;  it  is  bounded 
by  half  the  circumference  and  by  a  diameter. 

7th.  A  SEGMENT  of  a  circle  is  any  part  of  the  surface  cut  off 
by  a  right  line,  as  in  fig.  3.  Segments  may  be  therefore  greater 
or  less  than  a  semicircle. 

8th.  An  ARC  of  a  circle  is  any  portion  of  the  circumference  cut 
off,  as  C.  G.  D  or  E.  G.  F,  fig.  3. 

9th.  A  CHORD  is  a  right  line  joining  the  extremities  of  an  arc, 
as  C.  D  and  E.  F,  fig.  3.  The  diameter  is  the  chord  of  a 
semicircle.  The  chord  is  also  called  the  SUBTENSE. 

10th.  A  SECTOR  of  a  circle  is  a  space  contained  between  two 
radii  and  the  arc  which  they  intercept,  as  E.  C.  jD,  or  0.  C. 
H,  fig.  4. 

llth.  A  QUADRANT  is  a  sector  whose  area  is  equal  to  one-fourth 
of  the  circle,  as  fig.  5 ;  the.  arc  D.  E  being  equal  to  one-fourth 
of  the  whole  circumference,  and  the  radii  at  right  angles  to 
each  other. 

12.  A  DEGREE. — The  circumference  of  a  circle  is  considered  as 
divided  into  360  equal  parts  called  DEGREES,  (marked  °)  each 
degree  is  divided  into  60  minutes  (marked  ')  and  each  minute 
into  60  seconds  (marked  ");  thus  if  the  circle  be  large  or  small, 
the  number  of  divisions  is  always  the  same,  a  degree  being  equal 
to  1 -360th  part  of  the  whole  circumference,  the  semicircle  equal 
to  180°,  and  the  quadrant  equal  to  90°.  The  radii  drawn  from 
the  centre  of  a  circle  to  the  extremities  of  a  quadrant  are  always 
at  right  angles  to  each  other;  a  right  angle  is  therefore  called  an 
angle  of  90°.  If  we  bisect  a  right  angle  by  a  right  line,  it  would 
divide  the  arc  of  the  quadrant  also  into  two  equal  parts,  each  part 
equal  to  one-eighth  of  the  whole  circumference  containing  45°; 
if  the  right  angle  were  divided  into  three  equal  parts  by  straight 
lines,  it  would  divide  the  arc  into  three  equal  parts,  each  containing 
30°.  Thus  the  degrees  of  the  circle  are  used  to  measure  angles, 
and  when  we  speak  of  an  angle  of  any  number  of  degrees,  it  is 
understood,  that  if  a  circle  with  any  length  of  radius,  be  struck 
with  one  foot  of  the  dividers  in  the  angular  point,  the  sides  of  the 
angle  will  intercept  a  portion  of  the  circle  equal  to  the  number  of 
degrees  given. 

NOTE. — This  division  of  the  circle  is  purely  arbitrary,  but  it  has  existed 


PLATE    III. 


13 


from  the  most  ancient  times  and  every  where.  During  the  revolutionary 
period  of  1789  in  France,  it  was  proposed  to  adopt  a  decimal  division,  by 
which  the  circumference  was  reckoned  at  400  grades;  but  this  method  was 
never  extensively  adopted  and  is  now  virtually  abandoned. 

13.  The  COMPLEMENT  of  an  Jirc  or  of  an  JLngle,  is  the  difference 
between  that  arc  or  angle  and  a  quadrant;  thus  E.  D  fig.  6  is  the 
complement  of  the  arc  D.  B,  and  E.  C.  D  the  complement  of 
the  angle  D.  C.  B. 

14.  The  SUPPLEMENT  of  an  Arc  or  of  an  Jingle,  is  the  difference 
between  that  arc  or  angle  and  a  semicircle ;  thus  D.  Jl  fig.  7,  is 
the  supplement  of  the  arc  D.  By  and  D.  C.  Jl  the  supplement 
of  the  angle  B.  C.  D. 

15.  A  TANGENT  is  a  right  line,  drawn  without  a  circle  touching  it 
only  at  one  point  as  B.  E  fig.  8;  the  point  where  it  touches  the 
circle  is  called  the  point  of  contact,  or  the  tangent  point. 

16.  A  SECANT  is  a  right  line  drawn  from  the  centre  of  a  circle 
cutting  its  circumference  and  prolonged  to  meet   a  tangent  as 

C.  E  fig.  8. 

NOTE. — SECANT  POINT  is  the  same  as  point  of  intersection,  being  the 
point  where  two  lines  cross  or  cut  each  other. 

17.  The  CO-TANGENT  of  an  arc  is  the  tangent  of  the  comple- 
ment of  that  arc,  as  H.  K  fig.  S. 

NOTE. — The  shaded  parts  in  these  diagrams  are  the  given  angles,  but 
if  in  fig.  8,  D.  C.  H  be  the  given  angle  and  D.  H  the  given  arc,  then  H.  K. 
would  be  the  tangent  and  B.  E  the  co-tangent. 

18.  The  SINE  of  an  arc  is  a  line  drawn  from  one  extremity,  per- 
pendicular to  a  radius  drawn  to  the  other  extremity  of  the  arc  as 

D.  F  fig.  9. 

19.  The  CO-SINE  of  an  arc  is  the  sine  of  the  complement  of  that 
arc  as  L.  D  fig.  10. 

20.  The  VERSED  SINE  of  an  arc  is  that  part  of  the  radius  inter- 
cepted between  the  sine  and  the  circumference  as  F.  B  fig.  9. 

21.  In  figure  11,  we  have  the  whole  of  the  foregoing  definitions 
illustrated  in  one  diagram.      C.  H—C.  D — C.  B  and  C.  Jl  are 
Radii;  Jl.  B   the  Diameter;    B.    C.  D  a  Sector;   B.   C.  H  a 
Quadrant.     Let   B.    C.    D   be    the   given    Jingle,   and   B.    D 
the  given  Jlrc,  then  B.  D  is  the  Chord,  D.  H  the  Complement, 
and  D.  Jl  the  Supplement  of  the  arc;  Z>.  C.  H  the  Complement 
and  D.  C.  JL  the  Supplement  of  the  given  angle;  B.  E  the  Tan- 
gent and  H.  K  the  Co-tangent,   C.  E  the  Secant  and   C.  K  the 
Co-secant,    F.  D  the  Sine,  L.  D   the    Co-sine  and   F.  B  the 
Versed  Sine. 


14 


PLATE  IV. 

TO  ERECT  OR  LET  FALL  A  PERPENDICULAR. 


PROBLEM  1.  FIGURE  1. 


To  bisect  the  right  line  A.  B  by  a  perpendicular. 

1st.  With  any  radius  greater  than  one  half  of  the  given  line, 
and  with  one  point  of  the  dividers  in  A  and  B  successively, 
draw  two  arcs  intersecting  each  other,  in  C  and  D. 

2nd.  Through  the  points  of  intersection  draw  C.  D,  which  is 
the  perpendicular  required. 


PROBLEM  2.     FIG.  2. 


From  the  point  D  in  the  line  E.   F  to  erect  a  perpendicular. 

1st.  With  one  foot  of  the  dividers  placed  in  the  given  point  D 
with  any  radius  less  than  one  half  of  the  line,  describe  an  arc, 
cutting  the  given  line  in  B  and  C. 

2nd.  From  the  points  B  and  C  with  any  radius  greater  than  B. 
D,  describe  two  arcs,  cutting  each  other  in  G. 

3rd.  From  the  point  of  intersection  draw  G.  Z),  which  is  the  per- 
pendicular required. 


PROBLEM  3.     FIG.  3. 

To  erect  a  perpendicular  when  the  point  D  is  at  or  near  the  end  of 

a  line. 

1st.  With  one  foot  of  the  dividers  in  the  given  point  D  with 
any  radius,  as  D.  E,  draw  an  indefinite  arc  G.  H. 

2nd.  With  the  same  radius  and  the  dividers  in  any  point  of  the 
arc,  as  E,  draw  the  arc  B.  D.  F,  cutting  the  line  C.  D  in  B. 

3rd.  From  the  point  B  through  E  draw  a  right  line,  cutting 
the  arc  in  F. 


UITI7BRSITT 


Tit  KlifiCT  OR  LET  FALL  A  PERPENDICULAR. 


////.  2. 


Fiq.  5. 


CONSTRUCTION  AND  DIVISION  OF  ANGLES. 


Fia 


Fiq.  6. 


Fiq.  5. 


"  kSOT 


PLATE    IV.  15 

4th.    From  F  draw  F.  D,  which  is  the  perpendicular  required. 

NOTE. — It  will  be  perceived  that  the  arc  B.  D.  F  is  a  semicircle,  and 
the  right  line  B.  F  a.  diameter ;  if  from  the  extremities  of  a  semicircle  right 
lines  be  drawn  to  any  point  in  the  curve,  the  angle  formed  by  them  will  be 
a  right  angle.  This  affords  a  ready  method  for  forming  a  "  square  corner," 
and  will  be  found  useful  on  many  occasions,  as  its  accuracy  may  be  de- 
pended on. 


PROBLEM  4.     FIG.  4. 

Another  method  of  erecting  a  perpendicular  when  at  or  near  the 

end  of  the  line. 

Continue   the  line  H.   D    toward  C,  and  proceed  as  in  problem 
2;  the  letters  of  reference  are  the  same. 


PROBLEM  5.     FIG.  5. 
From  the  point  D  to  let  fall  a  perpendicular  to  the  line  A.  B. 

1st.  With  any  radius  greater  than  D.  G  and  one  foot  of  the  com- 
passes in  D,  describe  an  arc  cutting  Ji.  B  in  E  and  F. 

2nd.  From  E  and  F  with  any  radius  greater  than  E.  G;  describe 
two  arcs  cutting  each  other  as  in  C. 

3rd.  From  D  draw  the  right  line  D.  C,  then  D.  G  is  the  per- 
pendicular required. 

PROBLEM  6.     FIG.  6. 
When  the  point  D  is  nearly  opposite  the  end  of  the  line. 

1st.  From  the  given  point  D,  draw  a  right  line  to  any  point  of  the 

line  Jl.  B  as   0. 

2nd.  Bisect  0.  D  by  problem  1;  in  E. 
3rd.  With  one  foot  of  the  compasses  in  E  with  a  radius  equal  to 

E.  D  or  E.  0  describe  an  arc  cutting  Jl.  B  in  F. 
4th.  Draw  D.  F  which  is  the  perpendicular  required. 

NOTE. — The  reader  will  perceive  that  we  have  arrived  at  the  same  result 
as  we  did  by  problem  3,  but  by  a  different  process,  the  right  angle  being- 
formed  within  a  semicircle. 


16  PLATE    IV. 

PROBLEM  7.     FIG.  7. 

Another  method  of  letting  fall  a  perpendicular  when  the  given  point 
D  is  nearly  opposite  the  end  of  the  line. 

1st.  With  any  radius  as  F.  D  and  one  foot  of  the  compasses  in 
the  line  Jl.  B  as  at  F,  draw  an  arc  D.  H.  C. 

2nd.  With  any  other  radius  as  E.  D  draw  another  arc  D.  K. 
C,  cutting  the  first  arc  in  C  and  D. 

3rd.  From  D  draw  D.  C,  then  D.  G  is  the  perpendicular  re- 
quired. 

NOTE. — The  points  E  and  F  from  which  the  arcs  are  drawn,  should  be 
as  far  apart  as  the  line  A.  B  will  admit  of,  as  the  exact  points  of  intersec- 
tion can  be  more  easily  found,  for  it  is  evident,  that  the  nearer  two  lines  cross 
each  other  at  a  right  angle,  the  finer  will  be  the  point  of  contact. 


PROBLEM  8.     FIG.  8. 


To  erect  a  perpendicular  at  D  the  end  of  the  line  C.  D.  with  a  scale 

of  equal  parts. 

1st.    From  any  scale  of  equal  parts  take  three  in  your  dividers, 

and  with  one  foot  in  D,  cut  the  line   C.  D  in  B. 
2nd.    From  the  same  scale  take  four  parts  in  your  dividers,  and 

with  one  foot  in  D  draw  an  indefinite  arc  toward  E. 
3rd.    With  a  radius  equal  to  five  of  the  same  parts,  and  one  foot 

of  the  dividers  in  By  cut  the  other  arc  in  E. 
4th.    From  E  draw  E.  D}  which  is  the  perpendicular  required. 
NOTE  1st.     If  four  parts  were  first  taken  in  the   dividers  and  laid   off  on 

the  line   C.  D,  then   three  parts   should  be  used  for  striking  the  indefinite 

arc,  at  v3,  and  the  five  parts  struck  from  the  point  C,  which  w^ould  give  the 

intersection  «#,  and  arrive  at  the  same  result. 
2nd.     On  referring  to  the  definitions  of  angles,  it  will  be  found  that  the  side 

of  a  right  angled  triangle  opposite  the   right  angle  is  called  the  Hypothe- 

nuse;  thus  the  line  E.  B  is  the  hypothenuse  of  the  triangle  E.  D.  B. 
3rd.     The  square  of  the  hypothenuse  of  a  right  angled  triangle  is   equal  to 

the  sum  of  the  squares  of  both  the  other  sides. 
4th.     The  square  of  a  number  is  the  product  of  that  number  multiplied  by 

itself. 
Example.     The  length  of  the  side  D.  E  is  4,  which  multiplied   by   4,  will 

give  for  its  square  16.    The  length  of  D.  B  is  3,  which  multiplied  by  3,  gives 

for  the  square  9.     The  products  of  the  two  sides  added  together  give  25. 

The  length  of  the  hypothenuse  is  5,  which  multiplied  by  5,  gives  also  25. 
5th.     The  results  will  always  be  the  same,  but  if  fractional  parts  are  used  in 


PLATE  IV.  17 

the  measures,  the  proof  is  not  so  obvious,  as  the  multiplication  would  be 

more  complicated. 
6th.     3,  4  and  5  are  the  least  whole  numbers  that  can  be  used  in  laying  down 

this  diagram,  but  any  multiple  of  these  numbers  may  be  used ;  thus,  if  we 

multiply  them  by  2,  it  would  give  6,  8  and  10 ;  if  by  3,  it  would  give  9,  12 

and  15 ;  if  by  4 — 12,  16  and  20,  and  so  on.     Tne  greater  the    distances 

employed,  other  things  being  equal,  the  greater  will  be  the  probable  accuracy 

of  the  result. 
7th.     We  have  used  a  scale  of  equal  parts    without  designating  the  unit  of 

measurement,  which  may  be  an  inch,  foot,  yard,  or  any  other  measure. 
8th.     As  this  problem  is  frequently  used  by  practical  men  in  laying  off  work, 

we  will  give  an  illustration. 
Example.     Suppose  the  line  C.  D  to  be  the  front  of  a  house,  and  it  is  desired 

to  lay  oif  the  side  at  right  angles  to  it  from  the  corner  D. 
1st.     Drive  in  a  small  stake  at  D,  put  the  ring  of  a  tape  measure  on  it  and 

lay  off  twelve  feet  toward  B. 
2nd.     With  a  distance  of  sixteen  feet,  the  ring  remaining  at  D,  trace  a  short 

circle  on  the  ground  at  E. 
3rd.     Remove  the  ring  to  I?,  and  with  a  distance  of  twenty  feet  cut  the  first 

circle  at  E. 
4th.     Stretch  a  line  from  D  to  E,  which  will  give  the  required  side  of  the 

building. 


PLATE  V. 

CONSTRUCTION  AND  DIVISION  OF  ANGLES. 
PROBLEM  9.     FIG.  1. 


The  length  of  the  sides  of  a  Triangle  A.  B.,  C.  D.  and  E.  F 
being  given,  to  construct  the  Triangle,  the  two  longest  sides  to  be 
joined  together  at  A. 

1st.  With  the  length  of  the  line   C.  D  for  a  radius  and  one  foot 

in   Jly   draw  an  arc  at   G. 
2nd.  With  the  length  of  the  line  E.  F  for  a  radius  and  one  foot 

in  B,  draw  an  arc  cutting  the  other  arc  at    G. 
3rd.  From  the  point  of  intersection  draw    G.  A  and  G.  B,  which 

complete  the  figure. 


13  PLATE    V. 

PROBLEM  10.     FIG.  2. 
To  construct  an  Jingle  at  K  equal  to  the  Angle  H. 

1st.  From  H  with  any  radius,  draw  an  arc  cutting  the  sides  of  the 

angle  as  at  M  JV*. 
2nd.  From   K  with  the  same  radius,  describe  an   indefinite  arc 

at  0. 

3rd.  Draw  K.  0  parallel  to  H.  M. 

4th.  Take  the  distance  from  M  to  N  and  apply  it  from  0  to  P. 
5th.  Through  P  draw  K.  P,  which  completes  the  figure. 


PROBLEM  11.     FIG.  3. 
To  Bisect  the  given  Jingle  Q  by  a  Right  Line. 

1st.  With  any  radius  and  one  foot  of  the  dividers  in  Q  draw  an 

arc  cutting  the  sides  of  the  angle  as  in  R  and  S. 
2nd.  With  the  same  or  any  other  radius,  greater  than  one  half 

R.  S,  from  the  points  S  and  R,  describe  two  arcs  cutting  each 

other,  as  at  T. 
3rd.  Draw  T.  Q,  which  divides  the  angle  equally. 

NOTZ. — This  problem  may  be  very  usefully  applied  by  workmen  on  many 
occasions.  Suppose,  for  example,  the  corner  Q  be  the  corner  of  a  room, 
and  a  washboard  or  cornice  has  to  be  fitted  around  it;  first,  apply  the  bevel 
to  the  angle  and  lay  it  down  on  a  piece  of  board,  bisect  the  angle  as  above, 
then  set  the  bevel  to  the  centreline,  and  you  have  the  exact  angle  for  cutting 
the  mitre.  This  rule  will  apply  equally  to  the  internal  or  external  angle. 
Most  good  practical  workmen  have  several  means  for  getting  the  cut  of  the 
mitre,  and  to  them  this  demonstration  will  appear  unnecessary,  but  I  have 
seen  many  men  make  sad  blunders,  for  want  of  knowing  this  simple  rule. 

PROBLEM  12.     FIG.  4. 
To  Trisect  a  Right  Angle. 

1st.  From  the  angular  point  Fwith  any  radius,  describe  an  arc 
cutting  the  sides  of  the  angle,  as  in  X  and  W. 

2nd.  With  the  same  radius  from  the  points  X  and  Wy  cut  the  arc 
in  Y  and  Z. 

3rd.  Draw  Y.  V  and  Z.  V,  which  will  divide  the  angle  as  re- 
quired. 


PLATE    V.  19 

PROBLEM  13.     FIG.  5. 
In  the  triangle  A.  B.  C,  to  describe  a  Circle  touching  all  its  sides. 

1st.    Bisect  two  of  the  angles  by  problem  11,  as  Jl  and   B,  the 

dividing  lines  will  cut  each  other  in  D,  then  D  is  the  centre  of 

the  circle. 
2nd.    From  D  let  fall  a  perpendicular  to  either  of  the  sides  as  at 

F,  then   D.  F  is  the  radius,  with  which  to  describe  the  circle 

from  the  point  D. 


PROBLEM  14.     FIG.  6. 

On  the  given  line  A.  B  to  construct  an  Equilateral  Triangle,  the 
line  A.  B  to  be  one  of  its  sides. 

1st   With  a  radius  equal  to  the  given  line  from  the  points  Jl  and 
By  draw  two  arcs  intersecting  each  other  in  C. 
2nd.    From  (7,  draw  C.  Jl  and   C.  B,  to  complete  the  figure. 


PLATE   VI. 

CONSTRUCTION    OF    POLYGONS. 


A    figure     of  3  sides  is  called  a  Trigon. 

"                 4      "  "           Tetragon. 

polygon            5      "  Pentagon. 

"                 6      «  «           Hexagon. 

"                 7      "  «           Heptagon. 

"                 8      «  «           Octagon. 

9      "  Enneagon  or  Nonagon. 

a               10      "  «           Decagon. 

«               II"  "          Undecagon. 

"               12      "  «          Dodecagon. 

1st.  When  the  sides  of  a  polygon  are  all  of  equal  length  and  all  the 
angles  are  equal,  it  is  called  a  regular  polygon ;  if  unequal;  it  is 
called  an  irregular  polygon. 


20 


PLATE    VI. 


2nd.  It  is  not  necessary  to  say  a  regular  Hexagon,  regular  Octa- 
gon, &c.;  as  when  either  of  those  figures  is  named,  it  is  always 
supposed  to  be  regular,  unless  otherwise  stated. 


PROBLEM  15.     FIG.   1. 

On  a  given  line  A.  B  to  construct  a  square  whose  side  shall  be  equal 

to  the  given  line. 

1st.    With  the  length  A.  B  for  a  radius  from  the  points  JL  and  B, 

describe  two  arcs  cutting  each  other  in  C. 
2nd.  Bisect  the  arc  C.  A  or  C.  B  in  D. 
3rd.  From  C,  with  a  radius  equal  to  C.  D,  cut  the  arc  B.  E  in  E 

and  the  arc  A.  F  in  F. 
4th.   Draw  JL.  E,  E.  F  and  F.  B,  which  complete  the  square. 


PROBLEM  16.     FIG.  2. 
In  the  given  square  G.  H.  K.  J,  to  inscribe  an  Octagon. 

1st.  Draw  the  diagonals  G.  K  and  H.  J,  intersecting  each  other 
in  P. 

2nd.  With  a  radius  equal  to  half  the  diagonal  from  the  corners 
G.  H.  K  and  J,  draw  arcs  cutting  the  sides  of  the  square  in  0. 
0.  0,  &c. 

3rd.  Draw  the  right  lines  0.  0.,  0.  0,  &c.,  and  they  will  com- 
plete the  octagon. 

This  mode  is  used  by  workmen  when  they  desire  to  make  a 
piece  of  wood  round  for  a  roller,  or  any  other  purpose ;  it  is  first 
made  square,  and  the  diagonals  drawn  across  the  end ;  the  dis- 
tance of  one-half  the  diagonal  is  then  set  off,  as  from  G  to  R  in 
the  diagram,  and  a  guage  set  from  H  to  R  which  run  on  all  the 
corners,  gives  the  lines  for  reducing  the  square  to  an  octagon ; 
the  corners  are  again  taken  off,  and  finally  finished  with  a  tool 
appropriate  to  the  purpose.  The  centre  of  each  face  of  the  octa- 
gon gives  a  line  in  the  circumference  of  the  circle,  running  the 
whole  length  of  the  piece;  and  as  there  are  eight  of  those  lines 
equidistant  from  each  other,  the  further  steps  in  the  process  are 
rendered  very  simple. 


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Plate  7. 


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Fin.  2. 


. 


PLATE    VI.  21 

PROBLEM  17.     FIG.  3. 

In  a  given  circle  to  inscribe  an  Equilateral  Triangle,  a  Hexagon 

and  Dodecagon. 

1st.  For  the  TRIANGLE,  with  the  radius  of  the  given  circle  from 
any  point  in  the  circumference,  as  at  JH}  describe  an  arc  cutting 
the  circle  in  B  and  C. 

2nd.  Draw  the  right  line  B.  C,  and  with  a  radius  equal  B.  C,  from 
the  points  B  and  C,  cut  the  circle  in  D. 

3rd.    Draw  D.  B  and  D.    C,  which  complete  the  triangle. 

4th.  For  the  HEXAGON,  take  the  radius  of  the  given  circle  and 
carry  it  round  on  the  circumference  'six  times,  it  will  give  the 
points  Jl.  B.  E.  D.  F.  C,  through  them,  draw  the  sides  of  the 
hexagon.  The  radius  of  a  circle  is  always  equal  to  the  side  of 
an  hexagon  inscribed  therein. 

5th.  For  the  DODECAGON,  bisect  the  arcs  between  the  points 
found  for  the  hexagon,  which  will  give  the  points  for  inscribing 
the  dodecagon. 

PROBLEM  18.     FIG.  4. 
In  a  given  Circle  to  inscribe  a  Square  and  an  Octagon. 

1st.  Draw  a  diameter  Jl.  B,  and  bisect  it  with  a  perpendicular 
by  problem  1,  giving  the  points  C.  D. 

2nd.'  From  the  points  Jl.  C.  B.  D,  draw  the  right  lines  forming 
the  sides  of  the  square  required. 

3rd.  For  the  OCTAGON,  bisect  the  sides  of  the  square  and  draw 
perpendiculars  to  the  circle,  or  bisect  the  arcs  between  the  points 
Jl.  C.  B.  D,  which  will  give  the  other  angular  points  of  the  re- 
quired octagon. 

PROBLEM  19.     FIG.  5. 

On  the  given  line  O.  P  to  construct  a  Pentagon,  O.  P  being  the 

length  of  the  side. 

1st.  With  the  length  of  the  line  0.  P  from  0,  describe  the  semi- 
circle P.  Q,  meeting  the  line  P.  0,  extended  in  Q. 

2nd.  Divide  the  semicircle  into  five  equal  parts  and  from  0  draw 
lines  through  the  divisions  1,  2  and  3. 


PLATE    VI. 


3rd.  With  the  length  of  the  given  side  from  P,  cut  0  1  in  S,  from  j 
S  cut  0  2,  in  R,  and  from  Q  cut  0  2  in  R;  connect  the  points  0. 
Q.  R.  S.  P  by  right  lines,  and  the  pentagon  will  be  complete. 

PROBLEM  20.     FIG.  6. 

On  the  given  line  A.  B  to  construct  a  Heptagon,  A.  B  being  the 

length  of  the  side. 

1st.    From  Jl  with  JL.  B  for  a  radius,  draw  the  semicircle  B.  H. 

2d.  Divide  the  semicircle  into  seven  equal  parts,  and  from  Jl  \ 
through  1,  2,  3,  4  and  5,  draw  indefinite  lines. 

3rd.  From  B  cut  the  line  A  1  in  C,  from  C  cut  Jl  2  in  D,  from  j 
G  cut  Jl  4  in  F,  and  from  F  cut  ^  3  in  E,  connect  the  points  by  | 
right  lines  to  complete  the  figure. 

Any  polygon  may  be  constructed  by  this  method.  The  rule 
is,  to  divide  the  semicircle  into  as  many  equal  parts  as  there 
are  sides  in  the  required  polygon,  draw  lines  through  all  the 
divisions  except  two,  and  proceed  as  above. 

Considerable  care  is  required  to  draw  these  figures  accurately, 
on  account  of  the  difficulty  of  finding  the  exact  points  of  inter- 
section They  should  be  practised  on  a  much  larger  scale. 


PLATE   VII. 

PROBLEM  21.     FIG.   1. 
To  find  the   Centre  of  a  Circle. 

1st.    Draw  any  chord,  as  Jl.  B,  and  bisect  it  by  a  perpendicular  E. 

D,  which  is  a  diameter  of  the  circle. 
2nd.    Bisect  the  perpendicular  E.  D  by  problem  1,  the  point  of 

intersection  is  the  centre  of  the  circle. 

FIGURE  2. 

Another  method  of  finding  the  Centre  of  a  Circle. 

1st.    Join  any  three  points  in  the  circumference  as  F.  G.  H. 
2nd.    Bisect  the  chords  F.  G  and   G.  H  by  perpendiculars,  their 
point  of  intersection  at  C  is  the  centre  required. 


PLATE  VII.  23 

PROBLEM  22.     FIG.  3. 

To  draw  a  Circle  through  any  three  points  not  in  a  straight  line} 

as  M.  N.  O. 

1st.  Connect  the  points  by  straight  lines,  which  will  be  chords  to 
the  required  circle. 

2nd.  Bisect  the  chords  by  perpendiculars,  their  point  of  inter- 
section at  C  is  the  centre  of  the  required  circle. 

3rd.  With  one  foot  of  the  dividers  at  C,  and  a  radius  equal  to 
C.  M,  C.  JV*;  or  (7.  0,  describe  the  circle. 

PROBLEM  23.     FIG.  4. 
To  find  the  Centre  for  describing  the  Segment  of  a  Circle. 

1st.    Let  P.  R  be  the  chord  of  the  segment,  and  P.  S  the  rise. 

2nd.  Draw  the  chords  P.  Q  and  Q.  R,  and  bisect  them  by  per- 
pendiculars; the  point  of  intersection  at  C,  is  the  centre  for 
describing  the  segment. 


PROBLEM  24.     FIG.  5. 


To  find  a  Right  Line  nearly  equal  to  an  Jlrc  of  a  Circle,  as  H.  I.  R 

1st.  Draw  the  chord  H.  K,  and  extend  it  indefinitely  toward  0. 
2nd.  Bisect  the  segment  in  /,  and  draw  the  chords  H.  I  and  /.  K. 
3rd.  With  one  foot  of  the  dividers  in  H}  and  a  radius  equal  to  H. 

I,  cut  H.  0  in  M}  then  with  the  same  radius,  and  one  foot  in  M, 

cut  H.  0  again  in  JV*.  • 

4th.    Divide  the  difference  K.  JV*  into  three  equal  parts,  and  extend 

one  of  them  toward  0,  then  will  the  right  line  H.  0  be  nearly 

equal  to  the  curved  line  H.  I.  K. 


PROBLEM  25.     FIG.  6. 
To  find  a  Right  Line  nearly  equal  to  the  Semicircumference  A.  F.  B. 

1st.    Draw  the  diameter  Jl.  B,  and  bisect  it  by  the  perpendicular 

F.  H;  extend  F.  H  indefinitely  toward  G. 
2nd.    Divide  the  radius  C.  H  into  four  equal  parts,  and  extend 

three  of  those  parts  to   G. 
3rd.    At  F  draw  an  indefinite  right  line  D.  E,  parallel  to  A.  £. 


24 


PLATE  VII. 


4th.  From  G  through  JL,  the  end  of  the  diameter  Jl.  B,  draw  G. 
Jl.  D,  cutting  the  line  D.  E  in  D,  and  from  G  through  B  draw 
G.  B.  E,  cutting  D.  E  in  £,  then  will  the  line  D.  E  be  nearly 
equal  to  the  semicircumference  of  the  circle,  and  the  triangles 
D.  G.  E  and  Jl.  G.  B  will  be  equilateral. 

NOTE.— The  right  lines  found  by  problems  24  and  25,  are  not  mathemati- 
cally equal  to  the  respective  curves,  but  are  sufficiently  correct  for  all 
practical  purposes.  Workmen  are  in  the  habit  of  using  the  following 
method  for  finding  the  length  of  a  curved  line  : — 

They  open  their  compasses  to  a  small  distance,  and  commencing  at  one 
end,  step  off  the  whole  curve,  noting  the  number  of  steps  required,  and  the 
remainder  less  than  a  step,  if  any ;  they  then  step  off  the  same  number  of 
times,  with  the  same  distance,on  the  article  to  be  bent  around  it,  and  add 
the  remainder,  which  gives  them  a  length  sufficiently  true  for  their  purpose : 
the  error  in  this  method  amounts  to  the  sum  of  the  differences  between  the 
arc  cut  off  by  each  step,  and  its  chord. 


PLATE  VIII. 

PARALLEL    RULER    AND    APPLICATION. 

FIGURE   1. 

The  parallel  ruler  figured  in  the  plate  consists  of  two  bars  of  wood 
or  metal  Jl.  B  and  C.  D,  of  equal  length,  breadth  and  thickness, 
connected  together  by  two  arms  of  equal  length  placed  diagonal- 
ly across  the  bars,  both  at  the  same  angle,  and  moving  freely  on 
the  rivets  which  connect  them  to  the  bars;  if  the  bar  JL.  B  be 
kept  firmly  in  any  position  and  the  bar  C.  D  moved,  the  ends  of 
the  arms  connected  to  C.  D  will  describe  arcs  of  circles  and 
recede  from  JL.  B  until  the  arms  are  at  right  angles  to  the  bars, 
as  shewn  by  the  dotted  lines;  if  moved  farther  round,  the  bars  will 
again  approach  each  other  on  the  other  side. 

The  bars  of  which  the  ruler  is  composed,  being  parallel  to  each 
other.,  it  follows,  that  if  either  edge  of  the  instrument  is  placed 
parallel  with  a  line  and  held  in  that  position,  another  line  may  be 
drawn  parallel  to  the  first  at  any  distance  within  the  range  of  the 
instrument.  This  is  its  most  obvious  use;  it  is  generally  applied 
to  the  drawing  of  inclined  parallel  lines  in  mechanical  drawings, 
vertical  and  horizontal  lines  being  more  easily  drawn  with  the 
^quare,  when  the  drawing  is  attached  to  a  drawing  board. 


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PLATE  VIII.  25 

APPLICATION. — PROBLEM  26.     FIG.  2. 


To  divide  the  Line  E.  F  into  any  number  of  equal  parts,  say  12. 

1st.  From  E  draw  E.  G  at  any  angle  to  E.  F,  and  step  off  with 
any  opening  of  the  dividers  twelve  equal  spaces  on  E.  G. 

2nd.  Join  F  12,  and  with  the  parallel  ruler  draw  lines  through 
the  points  of  division  in  E.  G,  parallel  to  12  F,  intersecting  E. 
F,  and  dividing  it  as  required. 


PROBLEM  27.     FIG.  3. 

To  divide  a  Line  of  the  length  of  G.  H  in  the  same  proportion 
as  the  Line  I.  K  is  divided. 

1st.  From  7  draw  aline  at  any  angle  and  make  7.  L  equal  to  G.  H. 

2nd.  Join  the  ends  K  and  L  by  a  right  line,  and  draw  lines  par- 
allel to  it  through  all  the  points  of  division,  to  intersect  7.  L,  then 
7.  L  will  be  divided  in  the  same  proportion  as  7.  K. 


PROBLEM  28.     FIG.  4. 
To  reduce  the  Trapezium  A.  B.  C.  D  to  a  Triangle  of  equal  area. 

1st.    Prolong  C.  D  indefinitely. 

2nd.  Draw  the  diagonal  JL.  7),  place  one  edge  of  the  ruler  on  the 
line  A.  D  and  extend  the  other  edge  to  B,  then  draw  B.  E,  cut- 
ting C.  D  extended  in  E. 

3rd.  Join  JL.  E,  then  the  area  of  the  triangle  Ji.  E.  C  will  be  equal 
to  the  trapezium  Ji.  B.  C.  D. 

PROBLEM  29.     FIG.  5. 

To  reduce  the  irregular  Pentagon  F.  G.  H.  I.  K.  to  a  Tetragon 
and  to  a  Triangle,  each  of  equal  area  with  the  Pentagon. 

1st.  Prolong  7.  77  indefinitely. 

2nd.  Draw  the  diagonal  F.  77 and  G.  M  parallel  to  it,  cutting  7.  H 

in  J17,  and  draw  F.  M. 
3rd.  Prolong  K.  I  indefinitely  toward  L. 
4th.  Draw  the  diagonal  F.  I  and  draw  M.  L  parallel  to  it,  cutting 

K.  L  in  L. 


26  PLATE    IX. 

5th.  Draw  F.  L,  then  the  triangle  F.  L.  K,  and  the  tetragon  K. 
F.  M.  I,  are  equal  in  area  to  the  given  pentagon. 


PLATE   IX.- 

CONSTRUCTION  OF  THE  SCALE  OF  CHORDS  AND  ITS  APPLI- 
CATION.—PLANE  SCALES. 


PROBLEM  30.     FIG.   I. 

To   Construct  a  Scale  of  Chords. 

Let  Ji.  B  be  a  rule  on  which  to  construct  the  scale. 

1st.    With  any  radius,  and  one  foot  in  Dy  describe  a  quadrant; 

then  draw  the  radii  D.  C  and  D.  E. 
2nd.    Divide  the  arc  into  three  equal  parts  as  follows  : — With  the 

radius  of  the  quadrant,  and  the  dividers  in  C,  cut  the  arc  in  60 ; 

then,  with  one  foot  in  E,  cut  the  arc  in  30. 
3rd.    Divide  these  spaces  each  into  three  equal  parts,  when  the 

quadrant  will  be  divided  into  nine  equal  parts  of  10°  each. 
4th.    From  C,  draw  chords  to  each  of  the  divisions,  and  transfer 

them,  as  shewn  by  the  dotted  lines,  to  JL.  B. 
5th.    Divide  each  of  the  spaces   on   the  arc  into  ten  equal  parts, 

and  transfer  the  chords  to  Ji.  B,  when  we  shall  have  a  scale  of 

chords  corresponding  to  the  respective  degrees. 

NOTE  1. — This  scale  is  generally  found  on  the  plane  scale  which  accom- 
panies a  set  of  drawing  instruments,  and  marked  C,  or  Ch. 

NOTE  2. — Any  scale  of  chords  may  be  reconstructed  by  using  the  chord  of 
60°  as  a  radius  for  describing  the  quadrant. 


APPLICATION. — PROBLEM  31.     FIG.  2. 

To  lay  down  an  Jingle  at  F,  of  any  number  of  degrees,  say  25, 
the  line  G.  F  to  form  one  side  of  the  Jingle. 

1st.    Take  the  chord  of  60°  in  the  dividers,  and  with  one  foot  in 
F,  describe  an  indefinite  arc,  cutting  G.  F  in  H. 


PLATE    IX.  27 

2nd.    From  the  scale  take  25°  in  the  dividers,  and  wi  .h  one  foot 

in  H,  cut  the  arc  in  K. 
3rd.    Through  K,  draw  K.  Fy  which  completes  the  required  angle, 

If  we  desire  an  angle  of  ,15°  or  30°,  take  the  required  number 

from  the  scale,  and  cut  the  arc  in   0    or   P,  and  in  the  same 

manner  for  any  other  angle. 

PROBLEM  32.     FIG.  2. 

I 

To  measure  an  Jingle,  F,  already  Laid  down. 

1st.  With  the  chord  of  60°,  and  one  foot  of  the  dividers  in  the 
angular  point,  cut  the  sides  of  the  angle  in  H  and  K. 

2nd.  Take  the  distance  H.  K  in  the  dividers,  and  apply  it  to  the 
scale,  which  will  shew  the  number  of  degrees  subtended  by  the 
angle. 


SCALES  OF  EQUAL  PARTS. 


1st.  Scales  of  equal  parts  may  be  divided  into  two  kinds,  viz: — 
Those  which  consist  of  two  or  three  lines,  divided  by  short 
parallel  lines,  at  right  angles  to  the  others,  like  fig.  3,  or  those 
which  are  composed  of  several  parallel  lines,  divided  by  diagonal 
and  vertical  lines,  like  figs.  4  and  5 :  the  first  kind  are  called 
simple  scales,  the  second  diagonal  scales. 

2nd.  Scales  of  equal  parts  may  be  made  of  any  size,  and  may  be 
made  to  represent  any  unit  of  measure  :  thus  each  part  of  a  scale 
may  be  an  inch,  or  the  tenth  of  an  inch,  or  any  other  space,  and 
may  represent  an  inch,  foot,  yard,  fathom,  mile,  or  degree,  or  any 
other  quantity. 

3rd.  The  measure  which  the  scale  is  intended  to  represent,  is 
called  the  unit  of  measurement.  In  architectural  or  mechanical 
drawings,  the  unit  of  measurement  is  generally  a  foot,  which  is 
subdivided  into  inches  to  correspond  with  the  common  foot  rule. 
For  working  drawings,  the  scale  is  generally  large.  A  very 
common  mode  of  laying  down  working  drawings,  is  to  use  a 
scale  of  one  and  a  half  inches  to  the  foot ;  this  gives  one-eighth 
of  an  inch  to  an  inch,  which  is  equal  to  one-eighth  of  the  full 
size.  This  is  very  convenient,  as  every  workman  has  a  scale  on 


28  PLATE    IX. 

his  rule,  which  he  can  apply  to  the  drawing  with  facility.  Scales 
are  generally  made  to  suit  each  particular  case,  dependant  on  the 
size  of  the  object  to  be  represented,  and  on  the  size  of  the  paper 
or  board  on  which  the  drawing  is  to  be  made. 

4th.  To  DRAW  A  SCALE.  If  we  have  a  definite  size  for  each 
part  of  a  required  scale,  say  one-quarter  of  an  inch,  we  have  only 
to  extend  the  dividers  to  that  measure,  step  off  the  parts  and 
number  them,  reserving  the  left  hand  space  to  be  subdivided  for 
inches.  Care  should  be  taken  to  have  the  scale  true.  It  should 
be  proved  by  taking  two,  three,  four  or  five  parts  in  the  dividers, 
and  applying  it  to  several  parts  of  the  scale ;  when,  if  found 
correct,  the  drawing  may  be  proceeded  with.  It  is  much  easier 
to  draw  another  scale  if  the  first  is  imperfect,  than  to  correct  a 
drawing  made  from  a  false  scale. 

5th.  Fig.  3  requires  but  little  explanation.  It  is  called  a  quarter 
of  an  inch  scale,  as  each  unit  of  the  scale  is  a  quarter  of  an  inch ; 
the  starting  point  of  a  scale  marked  0,  is  called  its  ZERO.  The 
term  is  |iot  very  common  among  practical  men,  except  when 
applied  to  the  thermometer ;  and,  when  we  say  the  thermometer 
is  down  to  Zero,  we  mean  that  it  is  at  the  commencement  of  the 
scale.  It  is  better  to  number  a  scale  above  and  below,  as  in  the 
figure ;  for,  if  we  wish  to  take  a  measure  of  any  number  of  feet 
less  than  10,  and  inches,  say  3  feet  6  inches,  we  place  one  foot 
of  the  dividers  at  3,  numbered  from  below,  arid  extend  the  other 
out  to  6  inches.  If,  on  the  other  hand,  we  have  to  measure  a 
number  of  feet  more  than  10,  say  13,  we  should  place  one  foot 
in  the  division  marked  10,  on  the  top  of  the  scale  without  the 
plate,  and  extend  it  to  3,  when  we  are  enabled  to  read  the 
quantity  at  sight,  without  any  mental  operation,  as  we  must  do  if 
the  scale  is  only  numbered  as  below.  For  example,  to  take  13 
feet,  we  must  place  one  foot  of  the  dividers  at  20,  and  extend  it 
to  7.  This  operation  is  simple,  it  is  true,  but  it  requires  us  to 
subtract  7  from  20  to  get  13,  instead  of  reading  from  the  scale 
as  we  could  do  from  the  upper  numbers.  In  taking  a  large  space 
in  the  dividers,  it  is  always  better  to  take  the  whole  numbers 
first,  and  add  the  inches  or  other  fractions  afterwards  The  space 
on  the  left  hand  in  this  figure,  is  divided  into  twelve  parts  for 
inches. 


PLATE    IX.  29 

FIGURE  4. 

Is  a  HALF  INCH  DIAGONAL  SCALE,  divided  for  feet  and  inches. 
To  draw  a  scale  of  this  kind  : — 

1st.    Draw  7  lines  parallel  to  each  other,  and  equidistant. 

2nd.  Step  off  spaces  of  half  an  inch  each,  and  draw  lines  through 
the  divisions  across  the  whole  of  the  spaces. 

3rd.  Divide  the  top  of  the  first  space  into  two  equal  parts,  draw 
the  diagonal  lines,  and  number  them  as  in  the  diagram. 

To  take  any  measure  from  this  scale,  say  2'  1",  we  must  place  the 
dividers  on  the  first  line  above  the  bottom  on  the  second  division 
on  the*  scale,  and  extend  the  other  foot  to  the  first  diagonal  line, 
numbered  1,  which  will  give  the  required  dimension.  If  we 
wish  to  take  2'  1 17/,  we  must  place  one  leg  of  the  dividers  the 
same  as  before,  and  extend  the  other  to  the  second  diagonal  line, 
which  gives  the  dimension.  If  we  wish  to  take  1'  3",  we  must 
place  the  dividers  on  the  middle  line  in  the  first  vertical  division, 
and  extend  it  to  the  first  diagonal  line,  numbered  3,  and  proceed 
in  the  same  manner  for  any  other  dimension. 

This  is  a  very  useful  form  of  scale.  The  student  should  familiarize 
himself  with  its  construction  and  application. 


FIGURE  5. 

Is  AN  INCH  DIAGONAL  SCALE,  divided  into  tenths  and  hundredths. 

It  is  made  by  drawing  eleven  lines  parallel  to  each  other,  enclosing 
ten  equal  spaces,  with  vertical  lines  drawn  through  the  points  of 
division  across  the  whole.  The  left  hand  vertical  space  is  divided 
into  ten  equal  parts,  and  diagonal  lines  drawn  as  in  the  figure. 

This  scale  gives  three  denominations.  Each  of  the  small  spaces  on 
the  top  and  bottom  lines,  is  equal  to  one-tenth  of  the  whole  divi- 
sion. The  horizontal  lines  contained  between  the  first  diagonal 
and  the  vertical  line,  are  divided  into  tenths  of  the  smaller  divi- 
sion, or  hundreths  of  the  larger  division ;  for  example,  the  first 
line  from  the  top  contains  nine-tenths  of  the  smaller  division,  the 
second  eight-tenths,  the  third  seven-tenths,  and  so  on  as  num- 
bered on  the  end  of  the  scale.  To  make  a  diagonal  scale  of  this 
form,  divided  into  feet,  inches,  &,c.,  we  must  draw  13  parallel 
horizontal  lines,  and  divide  the  left  hand  space  also  into  13. 


30 


PLATE  X. 

CONSTRUCTION  OF  THE  PROTRACTOR. 


FIGURE  1. 


The  protractor  is  an  instrument  generally  formed  of  a  semicircle  and 
its  chord;  the  semicircle  is  divided  into  180  equal  parts  or  degrees, 
numbered  in  both  directions  from  10°  to  180°,  as  in  its  applica- 
tion, angles  are  often  required  to  be  measured  or  laid  down  on 
either  hand;  in  portable  cases  of  instruments  the  protractor  is 
frequently  drawn  on  a  flat  straight  scale  as  in  the  diagram.  Its 
mode  of  construction  is  sufficiently  obvious  from  the  drawing;  a 
small  notch  or  mark  in  the  centre  of  the  straight  edge  of  the  in- 
strument denotes  the  centre  from  which  the  semicircle  is  describ- 
ed, and  the  angular  point  in  which  all  the  lines  meet. 

APPLICATION. — PROBLEM  33.     FIG.  2. 


WITH  THE  PROTRACTOR,  TO  PROTRACT  OR  LAY  DOWN 

ANY  ANGLE. 

From  the  point  0  let  it  be  required  to  form  a  Right  Jingle  to  the 

line  O.  P. 

1st.  Place  the  straight  edge  of  the  protractor  to  coincide  with  the 
line  0.  P,  with  the  centre  at  0,  then  mark  the  angle  of  90°  at  S. 

2nd.  From  S  draw  S.  0,  which  gives  the  required  angle. 

While  the  instrument  is  in  the  position  described,  with  its  centre 
at  0,  any  other  angle  may  be  laid  down,  thus  at  Q  we  have  30°, 
at  R  60°,  at  T  120°,  and  at  V  150°,  and  so  on  from  the  fraction 
of  a  degree  up  to  180.° 

The  protractor  may  also  be  used  for  constructing  any  regular  po- 
lygon in  a  circle  or  on  a  given  line ;  to  do  so,  it  is  necessary  to 
know  the  angle  formed  by  said  polygon  by  lines  drawn  from  its 
corners  to  the  centre  of  the  circle,  and  also  to  know  the  angle 
formed  by  any  two  adjoining  faces  of  the  polygon.  The  table 
given  for  this  purpose  is  constructed  as  follows : 

1st.  To  find  the  angle  formed  by  any  polygon  at  the  centre,  divide 
360,  the  number  of  degrees  in  the  whole  circle,  by  the  number  of 


— 


flak*  Ki 
PROTRACTOR 


Its  Construction  ami  . 


Trie/on 

Ti'  I  rflf_IOTl  •  . 

Pentaon 


Octaon 


jD^ft  if  /on 
Dodecagon 


/Im/les  of  Regular  Polygons 

1XO  at  flic  I'liiltr  ()O     <it  ///c  i'in-n mfci'eiice 

HO  d" 

72  ...d° 

60     d? 

45  „&?....  /.">.' 

w  ...a?....  i4o 

36    d?. M4 

3O  .-£?-...  k~>t> 


FLAT  SEGMU'XTS  . I XI)   /', / It. Ul  f>LAS. 


^:R^:§ 

OF  THT 

SITY 


PLATE    X.  31 

sides  in  the  required  polygon,  the  quotient  will  be  the  angle  at 
the  centre;  for  example,  let  it  be  required  to  find  the  angle  at  the 
centre  of  an  octagon: — divide  360  by  8,  the  nilfnber  of  sides,  the 
quotient  will  be  45,  which  is  the  angle  formed  by  the  octagon  at 
the  centre. 

2nd.  To  find  the  angle  formed  by  two  adjoining  faces  of  a  polygon, 
we  must  subtract  from  180  the  number  of  degrees  in  the  semi- 
circle, the  angle  formed  by  said  polygon  at  the  centre,  the  re- 
mainder will  be  the  angle  formed  at  the  circumference.  For  ex- 
ample let  us  take  the  octagon ;  we  have  found  in  the  last  paragraph 
that  the  angle  formed  by  that  figure  in  the  centre  is  45°;  then  if 
we  subtract  45  from  180  it  will  leave  135,  which  is  the  angle 
formed  by  two  adjoining  faces  of  the  octagon. 


PLATE   XT. 

TO  DESCRIBE  FLAT  SEGMENTS  OF  CIRCLES  AND  PARABOLAS, 


Very  often  in  practice  it  would  be  very  inconvenient  to  find  the 
centre  for  describing  a  flat  segment  of  a  circle,  in  consequence  of 
the  rise  of  an  arch  being  so  small  compared  to  its  span. 


PROBLEM  34.     FIG.   1. 

To  describe  a  Segment  with  a  Triangle. 

NOTE. — In  all  the  diagrams  in  this  plate  Ji.  B  is  the  span  of  the  arch,  A.  D 
the  rise,  and  C  the  centre  of  the  crown  of  the  arch. 

1st.  Make  a  triangle  with  its  longest  side  equal  to  the  chord  or 
span  of  the  arch  and  its  height  equal  to  one-half  the  rise. 

2nd.  -Stick  a  nail  at  JL  and  at  C,  pi  ace  the  triangle  as  in  the  diagram 
and  move  it  round  against  the  nails  toward  JL*  a  pencil  kept  at  the 
apex  of  the  triangle  will  describe  one-half  of  the  curve. 

3rd.  Stick  another  nail  at  B,  and  with  the  triangle  moving  against 
C  and  B  describe  the  other  half  of  the  curve. 


32  PLATE    XI. 

PROBLEM  35.     FIG.  2. 


To    describe   the   same    Curve   tvith   strips   of  wood,  forming  a 

Triangle. 

1  st.  Drive  a  nail  at  A  and  another  at  J5,  place  one  strip  against  JL 

and  bring  it  up  to  the  centre  of  the  crown  at  C. 
2nd.  Place  another  strip  against  B  and  crossing  the  first  at  C,  nail 

them  together  at  the  intersection,  and  nail  a  brace  across  to  keep 

them  in  position. 
3rd.  With  the  pencil  at   C  and  the  triangle  formed  by  the  strips 

kept  against  Jl  and  B,  describe  the  curve  from  C  toward  Jl,  and 

from  C  toward  B. 


PROBLEM  36.     FIG.  3. 

To  draw  a  Parabolic   Curve  by  the  intersection  of  lines  forming 
Tangents  to  the   Curve. 

1st.  Draw  C.  8  perpendicular  to  Jl.  B,  and  make  it  equal  to 
Jl.  D 

2nd.  Join  Ji.  8  and  B.  8,  and  divide  both  lines  into  the  same 
number  of  equal  parts,  say  8,  number  them  as  in  the  figure,  draw 
1.  1. — 2.  2. — 3.  3.,  &c.,  then  these  lines  will  be  tangents  to  the 
curve ;  trace  the  curve  to  touch  the  centre  oi'  each  of  those  lines 
between  the  points  of  intersection. 


PROBLEM  37.     FIG.  4. 
To  draw  the  same   Curve  by  another  method. 

1st.    Divide  Jl.  D  and  B.  E,  into  any  number  of  equal  parts,  and 

C.  D  and  C.  E  into  a  similar  number. 
2nd.    Draw  1.  1. — 2.  2.  &c.,  parallel  to  Jl.  D,  and  from  the  points 

of  division  in  Jl.  D  and  B.  E,  draw  lines  to  C.     The  points  of 

intersection  of  the  respective  lines,  are  points  in  the  curve. 
NOTE. — The  curves  found,  as  in  figs.  3  and  4,  are  quicker  at  the  crown  than 

a  true  circular  segment ;  but,  where  the  rise  of  the  arch  is  not  more  than 

one-tenth  of  the  span,  the  variation  cannot  be  perceived. 


Hate  12 


OVAL  h'K'.riiKX  rai//Y/S'/iV;  OF  AIICS  OP 


flatelS. 


CYCLOID  AXD  EPICYCLOID. 


PLATE  XI.  33 

PROBLEM  38.     FIG.  5. 

To  describe  a  True  Segment  of  a   Circle  by  Intersections. 

1st.  Draw  the  chords  Jl.  C  and  B.  C,  and  Jl.  0  and  B.  0',  per- 
pendicular to  them. 

2nd.  Prolong  D.  E  in  each  direction  to  O.  0' ;  divide  0.  C,  C. 
O'y  Jl.  Dy  Jl.  6,  B.  6,  and  B.  E  into  the  same  number  of  equal 
parts. 

3rd.    Join  the  points  1.  L— 2.  2.  &c.,  in  Jl.  B  and  0.  Of. 

4th.  From  the  divisions  in  Jl.  D,  and  B.  E,  draw  lines  to  C. 
The  points  of  intersection  of  these  lines  with  the  former,  are 
points  in  the  curve.  A  semicircle  may  be  described  by  this  method. 


PLATE    XII. 

TO  DESCRIBE  OVAL   FIGURES  COMPOSED  OF  ARCS  OF 

CIRCLES. 


PROBLEM  39.     FIG.  1. 

The  length  of  the   Oval  A.  B.  being  given,  to  describe  an    Oval 

upon  it. 

1st.    Divide  Jl.  B  the  given  length,  into  three  equal  parts,  in  E 

and  F. 
2nd.    With  one  of  those  parts  for  a  radius,  and  the  compasses  in 

E  and  F  successively,  describe  two  circles  cutting  each  other  in 

0  and  0'. 
3rd.    From   the  points  of  intersection  in  0  and   0',  draw  lines 

through  E  and  F,  cutting  the  circles  in  V.   V,"  and  V.'  VJ" 
4th.    With  one  foot  of  the  compasses  in  0,  and  0'  successively,  and 

with  a  radius  equal  to  0.  V,"  or  0.'  V,  describe  the  arcs  between 

V.  V}  and  V!1   V",  to  complete  the  figure. 


PROBLEM  40.     FIG.  2. 

To  describe  the  Oval,  the  length  A.  B,  and  breadth  C.  D,  being 

given. 

1st.    With  half  the  breadth  for  a  radius,  and  one  foot  in  F,  de- 
scribe the  arc  C.  E,  cutting  Jl.  B  in  E. 


34  PLATE    XII. 

2nd.    Divide  the  difference  E.  B  between  the  semiaxes  into  three 

equal  parts,  and  carry  one  of  those  divisions  toward  4. 
3d.    Take  the  distance  B  4,  and  set  off  on  each  side  of  the  centre 

F  at  H  and  H.1 
4th.    With  the  radius  H.  HJ  describe  from  H  and  H'  as  centres, 

arcs  cutting  each  other  in  K  and  KJ 
5th.    From  K  and  K]  through  H  and  JET/  draw  indefinite  right 

•lines. 
6th.    With  the  dividers  in  H,  and  the  radius  H.  Jl,  describe  the 

curve  V.  JL.  V"  and  with  the  dividers  in  H/  describe  the  curve 

VJ  B.  VJ" 
7th.    From  K  and  K,'  with  a  radius  equal  to  K.   C,  describe  the 

curves  V.  C.  VJ  and  VJ'  D.  VJ"  to  complete  the  figure. 


PROBLEM  41.     FIG.  3. 

Jlnother  method  for  describing    the   Oval,  the  length  A.  B,  and 
breadth  C.  D,  being  given. 

1st.  Draw  C.  B,  and  from  B,  with  half  the  transverse  axis,  B.  F, 
cut  B.  C  in  0. 

2nd.  Bisect  B.  0  b j  a  perpendicular,  cutting  Jl.  B  in  P,  and  (7. 
D  in  Q. 

3rd.  From  F,  set  off  the  distance  F.  P  to  R,  and  the  distance 
F.  Q  to  S. 

4th.  From  S,  through  R  and  P,  and  from  Q  through  7?,  draw 
indefinite  lines. 

5th.  From  P  and  R,  and  from  S  and  Q,  describe  the  arcs,  com- 
pleting the  figure  as  in  the  preceding  problem. 

NOTE. — In  all  these  diagrams,  the  result  is  nearly  the  same.  Figs  1  and  2 
are  similar  figures,  although  each  is  produced  by  a  different  process.  The 
proportions  of  an  oval,  drawn  as  figure  1,  must  always  be  the  same  as  in 
the  diagram  ;  but,  in  figs.  2  and  3,  the  proportions  may  be  varied ;  but, 
when  the  difference  in  the  length  of  the  axes,  exceeds  one-third  of  the  longer 
one,  the  curves  have  a  very  unsightly  appearance,  as  the  change  of  curva- 
ture is  too  abrupt.  These  figures  are  often  improperly  called  ellipses,  and 
sometimes  false  ellipses.  Ovals  are  frequently  used  for  bridges.  When 
the  arch  is  flat,  the  curve  is  described  from  more  than  two  centres,  but  it  is 
never  so  graceful  as  the  true  ellipsis. 


35 


PLATE    XIII. 

TO  DESCRIBE  THE  CYCLOID  AND  EPICYCLOID. 

The  Cycloid  is  a  curve  formed  by  a  point  in  the  circumference  of 
a  circle,  revolving  on  a  level  line ;  this  curve  is  described  by  any 
point  in  the  wheel  of  a  carriage  when  rolling  on  the  ground. 

PROBLEM  42.     FIG.   1. 

To  find  any  number  of  Points  in  the  Cycloid  Curve  by  the  inter- 
section of  lines. 

1st.  Let  G.  H  be  the  edge  of  a  straight  ruler,  and  C  the  centre 
of  the  generating  circle. 

2nd.  Through  C  draw  the  diameter  Jl.  B  perpendicular  to  G.  H, 
and  E.  F  parallel  to  G.  H;  then  A.  B  is  the  height  of  the  curve, 
and  E.  F  is  the  place  of  the  centre  of  the  generating  circle  at  every 
point  of  its  progress. 

3rd.  Divide  the  semicircumference  from  B  to  Jl  into  any  number 
of  equal  parts,  say  8,  and  from  Jl  draw  chords  to  the  points  of  di- 
vision. 

4th.  From  C,  with  a  space  in  the  dividers  equal  to  one  of  the  di- 
visions on  the  circle,  step  off  on  each  side  the  same  number  of 
spaces  as  the  semicircumference  is  divided  into,  and  through  the 
points  draw  perpendiculars  to  G.  H:  number  them  as  in  the  dia- 
gram. 

5th.  From  the  points  of  division  in  E.  F,  with  the  radius  of  the 
generating  circle,  describe  indefinite  arcs  as  shewn  by  the  dotted 
lines. 

6th.  Take  the  chord  Jl  1  in  the  dividers,  and  with  the  foot  at  1 
and  1  on  the  line  G.  H}  cut  the  indefinite  arcs  described  from  1 
and  1  respectively  at  D  and  Df,  then  D  and  D'  are  points  in  the 
curve. 

7th.  With  the  chord  Jl  2,  from  2  and  2  in  G.  H,  cut  the  indefinite 
arcs  in  J  and  J7,  with  the  chord  Jl  3,  from  3  and  3,  cut  the  arcs  in 
K  and  K  and  apply  the  other  chords  in  the  same  manner,  cutting 
the  arcs  in  L.  M,  &LC. 

8th.  Through  the  points  so  found  trace  the  curve. 


36 


PLATE    XIII. 


NOTE. — The  indefinite  arcs  in  the  diagram  represent  the  circle  at  that  point 
of  its  revolution,  and  the  points  D.  J.  K,  &c.,  the  position  of  the  genera- 
ting point  B  at  each  place.  This  curve  is  frequently  used  for  the  arches  of 
bridges,  its  proportions  are  always  constant,  viz :  the  span  is  equal  to  the 
circumference  of  the.  generating  circle  and  the  rise  equal  to  its  diameter. 
Cycloidal  arches  are  frequently  constructed  which  are  not  true  cycloids,  but 
approach  that  curve  in  a  greater  or  less  degree. 


FIGURE  2. — THE  EPICYCLOID. 


This  curve  is  formed  by  the  revolution  of  a  circle  around  a  circle, 
either  within  or  without  its  circumference.,  and  described  by  a 
point  B  in  the  circumference  of  the  revolving  circle.  P  is  the 
centre  of  the  revolving  circle,  and  Q  of  the  stationary  circle. 


PROBLEM  43. 


To  find  Points  in  the  Curve. 

1st.  Draw  the  diameter  8.  8,  and  from  Q  the  centre,  draw  Q.  B 
at  right  angles  to  8.  8. 

2nd.  With  the  distance  Q.  P  from  Q,  describe  an  arc  0.  0  repre- 
senting the  position  of  the  centre  P  throughout  its  entire  progress. 

3rd.  Divide  the  semicircle  B.  D  and  the  quadrants  D.  8  into  the 
same  number  of  equal  parts,  draw  chords  from  D  to  1,  2,  3,  &,c., 
and  from  Q  draw  lines  through  the  divisions  in  D.  8  to  intersect 
the  curve  0.  0  in  1,  2,  3,  &c. 

4th.  With  the  radius  of  P  from  1,  2,  3,  &c.,  in  0.  0  describe  in- 
definite arcs,  apply  the  chords  D  1,  D  2,  &,c.,  from  1,  2,  3,  &c., 
in  the  circumference  of  Q,  cutting  the  indefinite  arcs  in  Jl.  C.  E. 
F,  &,c.,  which  are  points  in  the  curve. 


PLATE  XIV- 

DEFINITIONS    OF    SOLIDS 


On  referring  back  to  our  definitions,  we  find  that  a  point  has  posi- 
tion without  magnitude. 

A  Line  has  length,  without  breadth  or  thickness,  consequently 
has  but  one  dimension. 


UNIVERSITY; 


15. 


SO/JDS  AXD  THE  IK  COVERINGS. 


^Hfe'  ^^^ 


Fig.  4. 


PLATE     XIV.  37 

A  Surface  has  length  and  breadth,  without  thickness,  conse- 
quently has  two  dimensions,  which,  multiplied  together,  give  the 
content  of  its  surface. 

A  Solid  has  length,  breadth  and  thickness.  These  three  dimen- 
sions multiplied  together,  give  its  solid  content. 

Lineal  Measure,  is  the  measure  of  lines. 

Superficial  or  Square  Measure,  the  measure  of  surfaces, 

Cubic  Measure,  is  the  measure  of  solids. 

For  Example. — If  we  take  a  cube  whose  edge  measures  two  feet, 
then  two  feet  is  the  lineal  dimension  of  that  line.  If  the  edge  is 
two  feet  long,  the  adjoining  edge  is  also  two  feet  long;  then, 
two  feet  multiplied  by  two,  gives  four  feet,  which  is  the  superfi- 
cial content  of  a  face  of  the  cube. 

Then,  if  we  multiply  the  square  or  superficial  content,  by  two 
feet,  which  is  the  thickness  of  the  cube,  it  will  give  eight  feet, 
which  is  its  solid  content. 

Then,  two  lineal  feet  is  the  length  of  the  edge. 
"      four  square  "     the  surface  of  one  side. 

And  eight  cubic      "     the  solid  content  of  the  cube. 


THE  CUBE  OR  HEXAHEDRON,  ITS  SECTIONS  AND  SURFACE. 


FIGURE  1. 

1st.  The  cube  is  one  of  the  regular  polyhedrons,  composed  of  six 
regular  square  faces,  and  bounded  by  twelve  lines  of  equal 
length ;  the  opposite  sides  are  all  parallel  to  each  other. 

2nd.  If  a  cube  be  cut  through  two  of  its  opposite  edges,  and  the 
diagonals  of  the  faces  connecting  them,  the  section  will  be  an 
oblong  rectangular  parallelogram,  as  fig.  2. 

3rd.  If  a  cube  be  cut  through  the  diagonals  of  three  adjoining 
faces,  as  in  fig  3,  the  section  will  be  an  equilateral  triangle,  whose 
side  is  equal  to  the  diagonal  of  a  face  of  the  cube.  Two  such 
sections  may  be  made  in  a  cube  by  cutting  it  again  through  the 
other  three  diagonals,  and  the  second  section  will  be  parallel  to 
the  first. 

4th.  If  a  cube  be  cut  by  a  plane  passing  through  all  its  sides,  the 
line  of  section,  in  each  face,  to  be  parallel  with  the  diagonal,  and 
midway  between  the  diagonal  and  the  corner  of  the  face,  as  in 


38  PLATE    XIV. 

fig.  4,  the  section  will  be  a  regular  hexagon,  and  will  be  parallel 
with,  and  exactly  midway  between  the  triangular  sections  de- 
fined in  the  last  paragraph. 

5th.  If  a  cube  be  cut  by  any  other  plane  passing  through  all  its 
sides,  the  section  will  be  an  irregular  hexagon. 

6th.  The  surface  of  the  cube  fig.  1,  is  shewn  at  fig.  5,  and  if  a 
piece  of  pasteboard  be  cut  out,  of  that  form,  and  cut  half  through 
in  the  lines  crossing  the  figure,  then  folded  together,  it  will 
form  the  regular  solid.  All  the  other  solids  may  be  made  of 
pasteboard,  in  the  same  manner,  if  cut  in  the  shape  shewn  in  the 
coverings  of  the  diagrams  in  the  following  plates. 

7th.  The  measure  of  the  surface  of  a  cube  is  six  times  the  square 
of  one  of  its  sides.  Thus,  if  the  side  of  a  cube  be  one  foot,  the 
surface  of  one  side  will  be  one  square  foot,  and  its  whole  surface 
would  be  six  square  feet. 

Its  solidity  would  be  one  cubic  foot. 

NOTE. — The  cube  may  also,  in  general,  be  called  a  prism,  and  a  parallelo- 
pipedon,  as  it  answers  the  description  given  of  those  bodies,  but  the  terms 
are  seldom  applied  to  it. 


PLATE    XV. 

SOLIDS     AND     THEIR     COVERINGS 


FIG.  1.  Is  a  solid,  bounded  by  six  rectangular  faces,  each  oppo- 
site pair  being  parallel,  and  equal  to  each  other ;  the  sides  are 
oblong  parallelograms,  and  the  ends  are  squares.  It  is  called  a 

right  SQUARE  PRISM,  PARALLELOPIPED,  Or  PARALLELOPIPEDON. 

FIG.  2.     Is  its  covering  stretched  out. 

FIG.  3.  Is  a  triangular  prism;  its  sides  are  rectangles,  and  its 
ends  equal  triangles. 

FIG.  4.     Is  its  covering. 

PRISMS  derive  their  names  from  the  shape  of  their  ends,  and  the 
angles  of  their  faces,  thus  :  Fig.  1  is  a  square  prism,  and  fig.  2 
a  triangular  prism.  If  the  ends  were  pentagons  the  prism 
would  be  pentagonal;  if  the  ends  were  hexagons,  the  prism 


PLATE    XV.  39 

would  be  hexagonal,  fyc.  The  sides  of  all  regular  prisms  are 
equal  rectangular  parallelograms. 

Fie.  5.  Is  a  SQUARE  PYRAMID,  bounded  by  a  square  at  its 
base,  and  four  regular  triangles,  as  shewn  at  fig.  6. 

Pyramids,  like  prisms,  derive  their  names  from  the  shape  of  their 
bases ;  thus  we  may  have  a  square  pyramid,  as  in  fig.  5,  or  a 
triangular,  pentagonal,  or  hexagonal  pyramid,  &c.,  as  the  base  is 
a  triangle,  pentagon,  hexagon,  or  any  other  figure. 

The  sides  of  a  pyramid  incline  together,  forming  a  point  at  the 
top.  This  point  is  called  its  vertex,  apex,  or  summit. 

The  axis  of  a  pyramid,  is  a  line  drawn  from  its  summit,  to  the 
centre  of  its  base.  *The  length  of  the  axis,  is  the  altitude  of  the 
pyramid.  When  the  base  of  a  pyramid  is  perpendicular  to  its 
axis,  it  is  called  a  right  pyramid;  if  they  are  not  perpendicular 
to  each  other,  the  pyramid  is  oblique.  If  the  top  of  a  pyramid  be 
cut  off,  the  lower  portion  is  said  to  be  truncated  ;  it  is  also  called 
a  frustrum  of  a  pyramid,  and  the  upper  portion  is  still  a  pyra- 
mid, although  only  a  segment  of  the  original  pyramid. 

A  pyramid  may  be  divided  into  several  truncated  pyramids,  or  frus- 
trums,  and  the  upper  portion  remain  a  pyramid,  as  the  name 
does  not  convey  any  idea  of  size,  but  a  definite  idea  of  form, 
viz  :*  a  solid,  bounded  by  an  indefinite  number  of  equal  triangles, 
with  their  edges  touching  each  other,  forming  a  point  at  the  top. 

A  pyramid  is  said  to  be  acute,  right  angled  or  obtuse,  dependant 
on  the  form  of  its  summit. 

An  OBELISK  is  a  pyramid  whose  height  is  very  great  compared 
to  the  breadth  of  the  base.  The  top  of  an  obelisk  is  generally 
truncated  and  cut  off,  so  as  to  form  a  small  pyramid,  resting  on 
the  frustrum,  which  forms  the  lower  part  of  the  obelisk. 

When  the  polygon,  forming  the  base  of  a  pyramid,  is  irregular, 
the  sides  of  the  figure  will  be  unequal,  and  the  pyramid  is  called 
an  irregular  pyramid. 

*  These  definitions  are  applied  to  pyramids  that  are  right  and  regular:  it  is 
not  necessary  to  say,  "aright,  regular  pyramid,"  as  when  a  pyramid  is 
named,  it  is  always  supposed  to  be  right  and  regular,  unless  otherwise  ex- 
pressed. 


40 


PLATE    XVI. 

SOLIDS    AND     THEIR     COVERINGS. 


FIG.   1.     Is  an  HEXAGONAL  PYRAMID;  and  fig.  2  its  covering. 

FIG.  3.  A  RIGHT  CYLINDER,  is  bounded  by  two  uniform  circles, 
parallel  to  each  other.  The  line  connecting  their  centres,  is 
called  the  axis.  The  sides  of  the  cylinder  is  one  uniform  surface, 
connecting  the  circumferences  of  the  circle,  and  everywhere 
equidistant  from  its  axis. 


PROBLEM  44.     FIG.  4. 


To  find  the  Length  of  the  Parallelogram  A.  B.  C.  D,  to  form  the 

Side  of  the   Cylinder. 

1st.    Draw  the  ends,  and  divide  one  of  them  into  any  number  of 

equal  parts,  say  twelve. 
2nd.    With  the  space  of  one  of  those  parts,  step  off   the  same 

number  on  Ji.  B,  which  will  give  the  breadth  of  the  covering 

to  bend  around  the  circles. 
FIG.  5.     Is  A  RIGHT  CONE;  its  base  is  a  circle,  its  sides  sloping 

equally  from  the  base   to  its  summit.      A  line  drawn  from  its 

summit  to  the  centre  of  the  base,  is  called  its  axis.     If  the  axis 

and  base  are  not  perpendicular  to  each  other,  it  forms  an  oblique, 

or  scalene  cone. 


PROBLEM  45.     FIG.  6. 


To  draw  the   Covering. 

1st.    With  a  radius  equal  to  the  sloping  height  of  the  cone,  from 

E,  describe  an  indefinite  arc,  and  draw  the  radius  E.  F. 
2nd.    Draw  the  circle  of  the  base,  and  divide  its  circumference 

into  any  number  of  equal  parts,  say  twelve. 
3rd.    With  one  of  those  parts  in  the  dividers,  step  off  from  F 

the  same  number  of  times  to  G,  then  draw  the  radius  E.   G,  to 

complete  the  figure. 


41 


PLATE   XVIX, 

COVERINGS    OF    SOLIDS. 

FIG.  1.     THE  SPHERE 

Is  a  solid  figure  presenting  a  circular  appearance  when  viewed  in 
any  direction ;  its  surface  is  every  where  equidistant  from  a  point 
within,  called  its  centre. 

1st.  It  may  be  formed  by  the  revolution  of  a  semicircle  around  its 
chord. 

2nd.  The  chord  around  which  it  revolves  is  called  the  axis,  the 
ends  of  the  axis  are  called  poles. 

3rd.  Any  line  passing  through  the  centre  of  a  sphere  to  opposite 
points,  is  called  a  diameter. 

4th.  Every  section  of  a  sphere  cut  by  a  plane  must  be  a  circle,  if 
the  section  pass  through  the  centre,  its  section  will  be  a  great  cir- 
cle of  the  sphere ;  any  other  section  gives  a  lesser  circle. 

5th.  When  a  sphere  is  cut  into  two  equal  parts  by  a  plane  passing 
through  its  centre,  each  part  is  called  a  hemisphere;  any  part  of 
a  sphere  less  than  a  hemisphere  is  called  a  segment;  this  term 
may  be  applied  to  the  larger  portion  as  well  as  to  the  smaller. 


PROBLEM  46.     FIG.  2. 
To  draw  the   Covering  of  the  Sphere. 

1st  Divide  the  circumference  into  twelve  equal  parts. 

2nd.  Step  off  on  the  line  Jl.  B  the  same  number  of  equal  parts, 
and  with  a  radius  of  nine  of  those  parts,  describe  arcs  through  the 
points  in  each  direction ;  these  arcs  will  intersect  each  other  in  the 
lines  C.  D  and  E*  F,  and  form  the  covering  of  the  sphere. 


FIGURE  3. 


Is  the  surface  of  a  regular  TETRAHEDRON,  it  is  bounded  by  four 
equal  equilateral  triangles. 


42  PLATE    XVII. 

FIGURE  4. 

The  regular  OCTAHEDRON  is  bounded  by  eight  equal  equilateral 
triangles. 


FIGURE  5. 


The  DODECAHEDRON  is  bounded  by  twelve  equal  pentagons. 


FIGURE  6. 

The   ICOSAHEDRON    is    bounded    by    twenty   equal    equilateral 

triangles. 
The  four  last  figures,  together  with  the  hexahedron  delineated  on 

Plate  14,  are  all  the  regular  polyhedrons.     All  the  faces  and  all 

the  solid  angles  of  each  figure  are  respectively  equal.  These  solids 

are  called  platoriic  figures. 


PLATE  XVIII. 

THE    CYLINDER    AND    ITS    SECTIONS. 


1st.  If  we  suppose  the  right  angled  parallelogram  Jl.  B.  C.  D, 
fig.  1,  to  revolve  around  the  side  A.  B,  it  would  describe  a  solid 
figure;  the  sides  Jl.  D  and  B.  C  would  describe  two  circles 
whose  diameters  would  be  equal  to  twice  the  length  of  the  re- 
volving sides;  the  side  C.  D  would  describe  a  uniform  surface  con- 
necting the  opposite  circles  together  throughout  their  whole  cir- 
cumference. The  solid  so  described  would  be  a  RIGHT  CYLINDER. 

2nd.  The  line  Jl.  B,  around  which  the  parallelogram  revolved,  is 
called  the  AXIS  of  the  cylinder,  and  as  it  connects  the  centres  of 
the  circles  forming  the  ends  of  the  cylinder,  it  is  every  where 
equidistant  from  its  sides. 

3rd.  If  the  ends  of  a  cylinder  be  not  at  right  angles  to  its  axis,  it  is 
called  an  OBLIQUE  CYLINDER. 

4th.  If  a  cylinder  be  cut  by  any  plane  parallel  to  its  axis,  the  sec- 
tion will  be  a  parallelogram,  as  E.  F.  G.  H,  fig.  1. 


PLATE    XVIII.  43 

5th.  If  a  cylinder  be  cut  by  any  plane  at  right  angles  to  its  axis, 
the  section  will  be  a  circle. 

6th.  If  a  cylinder  be  cut  by  any  plane  not  at  right  angles  to  its 
axis,  passing  through  its  opposite  sides,  as  at  K.  L  or  M.  JV,  fig. 
2,  the  section  will  be  an  ELLIPSIS,  of  which  the  line  of  section 
K.  L  or  M.  JV*  would  be  the  longest  diameter,  called  the  TRANS- 
VERSE or  MAJOR  DIAMETER,  and  the  diameter  of  the  cylinder 
C.  D  would  be  the  shortest  diameter,  called  the  CONJUGATE  or 

MINOR  DIAMETER. 


PROBLEM  47.     FIG.  3 

To  describe  an  Ellipsis  from  the   Cylinder  with   a  string  and 

two  pins. 

1st.  Draw  the  right  lines  JV*.  M  and  C.  D  at  right  angles  to  each 

other,  cutting  each  other  in  S. 
2nd.  Take  in  your  dividers  the  distance  P.  M  or  P.  N,  fig.  2, 

and  set  it  off  from  S  to  M  and  JV,  fig.  3,  which  will  make  M.  JV* 

equal  to  M.  JV,  fig.  2. 
3d.    From  Jl,  fig.  2,  take  A.  D  or  Jl.  C,  and  set  it  off  from  S  to 

C  and  7),  making  C.  D  equal  to  the  diameter  of  the  cylinder. 
4th.  With  a  distance  equal  to  S.  M  or  S.  JV  from  the  points  D 

and  C,  cut  the  transverse  diameter  in  E  and  F;  then  E  and  F 

are  the  FOCI  for  drawing  the  ellipsis. 
NOTE. — E  is  a  FOCUS,  and  F  is  a  FOCUS.     E  and  F  are  FOCI. 
5th.    In  the  foci,  stick  two  pins,  then  pass  a  string  around  them, 

and  tie  the  ends  together  at  C. 
6th.    Place  the  point  of  a  pencil  at  C,  and   keeping  the  string 

tight,  pass  it  around  and  describe  the  curve. 
NOTE. — The  sum  of  all  lines  drawn  from  the  foci,  to  any  point  in  the  curve, 

is  always  constant  and  equal  to  the  major  axis  :  thus,  the  length  of  the  lines 

E.  R,  and  F.  R,  added  together,  is  equal  to  the  length  of  E.  C,  and  F.  C, 

added  together,  or  to  two  lines  drawn  from  E  and  F,  to  any  other  point  in 

the  curve. 

7th.  Fig.  4  is  the  section  of  the  cylinder,  through  L.  K,  fig.  2, 
and  is  described  in  the  same  way  as  fig.  3.  The  letters  of  refer- 
ence are  the  same  in  both  diagrams,  except  that  the  transverse 
diameter  L.  K,  is  made  equal  to  the  line  of  section  L.  K,  in 
fig.  2. 


44  PLATE    XVIII. 

8th.  The  line  JV.  M,  fig.  3,  or  L.  K,  fig.  4,  is  called  the  TRANS- 
VERSE, or  MAJOR  AXIS,  (plural  AXES,)  and  the  line  C.  D,  its 
CONJUGATE,  or  MINOR  AXIS.  They  are  also  called  the  transverse 
and  conjugate  diameters,  as  above  defined.  The  transverse  axis 
is  the  longest  line  that  can  be  drawn  in  an  ellipsis. 

9th.  Any  line  passing  through  the  centre  S,  of  an  ellipsis,  and 
meeting  the  curve  at  both  extremities,  is  called  a  DIAMETER  : 
every  diameter  divides  the  ellipsis  into  two  equal  parts.  The 
CONJUGATE  of  any  diameter,  is  a  line  drawn  through  the  centre, 
terminated  by  the  curve,  parallel  to  a  tangent  of  the  curve  at  the 
vertex  of  the  said  diameter.  The  point  where  the  diameter 
meets  the  curve,  is  the  vertex  of  that  diameter. 

10th.  An  ORDINATE  to  any  diameter,  is  a  line  drawn  parallel  to 
its  conjugate,  and  terminated  by  the  curve  and  the  said  diameter. 
An  ABSCISSA  is  that  portion  of  a  diameter  intercepted  between 
its  vertex  and  ordinate.  Unless  otherwise  expressed,  ordinates 
are  in  general,  referred  to  the  axis,  and  taken  as  perpendicu- 
lar to  it.  Thus,  in  fig. '4,  X.  Y  is  the  ordinate  to,  and  L.  X 
and  K.  X,  the  abscissae  of  the  axis  K.  L. —  V.  W  is  the  ordinate 
to,  and  (7.  F",  and  D.  F,  the  abscissae  of  the  axis  C.  D. 


PLATE   XIX. 

THE    CONE    AND    ITS    SECTIONS. 

DEFINITIONS 

1st.  A  CONE  is  a  solid,  generated  by  the  revolution  of  a  right 

angled  triangle  about  one  of  its  sides. 
2nd.    If  both  legs  of  the  triangle  are  equal,  as  S.  JV  and  JV.  0, 

fig.  2,  it  would  generate  a  RIGHT  ANGLED  CONE  ;  the  angle  S 

being  a  right  angle. 
3rd.    If  the  stationary  side  of  the  triangle  be  longest,  as  M.  J 

the   cone  will  be  ACUTE,  and  if  shortest,  as  T.  JV,  it  will  be 

OBTUSE  angled. 
4th.    The  BASE  of  a  cone  is  a  circle,  from  which  the  sides  slope 

regularly  to  a  point,  which  is  called  its  VERTEX,  APEX,  or  SUMMIT. 


rill-:   ('YIJM)l'Ji  AM)  SKCTIUXS. 


/•/,/.  I. 


Fig. 


TUK  COM!  JM)  ITS  SUCTIONS, 


/•'/'</  I 


lw.2. 


B          A 


W^-Mvufie. 


PLATE    XIX.  45 

5th.  The  AXIS  of  a  cone,  is  a  line  passing  from  the  vertex  to  the 
centre  of  the  base,  as  M.  JV*;  figs.  1,  2,  3  and  4,  and  represents 
the  line  about  which  the  triangle  is  supposed  to  rotate. 

6th.  A  RIGHT  CONE.  When  the  axis  of  a  cone  is  perpendicular 
to  its  base,  it  is  called  a  right  cone  ;  if  they  are  not  perpendicular 
to  each  other,  it  is  called  an  OBLIQUE  CONE. 

7th.  If  a  cone  be  cut  by  a  plane  passing  through  its  vertex  to  the 
centre  of  its  base,  the  section  will  be  a  TRIANGLE. 

8th.  If  cut  by  a  plane,  parallel  to  its  base,  the  section  will  be  a 
CIRCLE,  as  at  E.  F,  fig.  1. 

9th.  If  the  upper  part  of  fig.  1  should  be  taken  away,  as  at  E. 
F,  the  lower  part  would  be  a  TRUNCATED  CONE,  or  FRUSTRUM, 
the  part  above  E.  F,  would  still  be  a  cone ;  and,  if  another  por- 
tion of  the  top  were  cut  off  from  it,  another  truncated  cone  would 
be  formed :  thus  a  cone  may  be  divided  into  several  truncated 
cones,  and  the  portion  taken  from  the  summit,  would  still  remain 
a  cone.  Similar  remarks  have  already  been  applied  to  the 
pyramid. 

10th.  If  a  cone  be  cut  by  any  plane  passing  through  its  opposite 
sides,  as  at  Jl.  B,  fig.  3,  the  section  will  be  an  ellipsis. 

llth.  If  a  cone  be  cut  by  a  plane,  parallel  with  one  of  its  sides, 
as  at  P.  Q.,  R.  S,  or  Rr.  Sf,  fig.  4,  the  section  will  be  a  PARA- 
BOLA. 

12th.  If  a  cone  be  cut  by  a  plane,  which,  if  continued,  would 
meet  the  opposite  cone,  as  through  C.  D,  fig.  4,  meeting  the  op- 
posite cone  at  0?  the  section  will  be  an  HYPERBOLA. 


PROBLEM  48.     FIG.  3. 
To  describe  the  Ellipsis  from  the  Cone. 

1st.    Let  fig.  3  represent  the  elevation  of  a  right  cone,  and  Jl.  B 

the  line  of  section. 
2nd.    Bisect  Jl.  B  in  C. 
3rd.    Through  C,  draw  E.  F  perpendicular  to  the  axis  M.  JV, 

cutting  the  axis  in  P. 
4th.    With  one  foot  of  the  dividers  in  P,  and  a  radius  equal  to 

P.  E,  or  P.  F,  describe  the  arc  E.  D.  F. 
5th.    From  C,  the  centre  of  the  line  of  section  Jl.  B,  draw  C.  D 

parallel  to  the  axis,  cutting  the  arc  E.  D.  F  in  D. 


46  PLATE  XIX 

6th.  Then  JL.  B  is  the  transverse  axis,  and  C.  D  its  semiconju- 
gate  of  an  ellipsis,  which  may  be  described  with  a  string,  as  ex- 
plained for  the  section  of  the  cylinder,  or  by  any  of  the  other 
methods  to  be  hereafter  described. 

NOTE. — A  section  of  the  cylinder,  as  well  as  of  a  cone,  passing  through  it 
opposite  sides,  is  always  an  ellipsis.     In  the  cone,  the  length  of  both  axes 
vary  with  every  section,  but  in  the  cylinder,  the  conjugate  axis  is  always 
equal  to  the  diameter  of  the  cylinder,  whatever  may  be  the  inclination  of  the 
line  of  section. 


PROBLEM  49.     FIG.  4. 

To  find  the  length  of  the  base  line  for  describing  the  other  sections. 

1st.  With'  one  foot  of  the  dividers  in  JV,  and  a  radius  equal  to 
JY.  T,  or  J\T.  V)  describe  a  semicircle,  equal  to  half  the  base  of 
the  cone. 

2nd.  From  C  and  P,  the  points  where  the  sections  intersect  the 
base,  draw  P.  Jl,  and  C.  B,  cutting  the  semicircle  in  Jl  and  B. 
Then  A.  P  is  one-half  the  base  of  the  parabola.,  and  C.  B  is 
one-half  the  base  of  the  hyperbola.  The  methods  for  describing 
these  curves,  are  shewn  in  Plates  20  and  21. 


PLATE  XX. 

TO  DESCRIBE  THE  ELLIPSIS  AND  HYPERBOLA. 

PROBLEM  50.     FIG.  1. 
To  find  Points  in  the  Curve  of  an  Ellipsis  by  Intersecting  Lines. 

Let  Jl.  B,  be  the  given  transverse  axis,  and   C.  D,  the  conjugate. 
1st.    Describe   the   parallelogram   L.  M.  JY.  0,  the   boundaries 

passing  through  the  ends  of  the  axes. 
2nd.    Divide  Jl.  L,—J1.  J\T,—B.  M,  and  B.  0,  into  any  number 

of  equal  parts,  say  4,  and  number  them  as  in  the  diagram. 
3rd.    Divide  JL.  $,  and  B.  S,  also  into  4  equal  parts,  and  number 

them  from  the  ends  toward  the  centre. 


PLATE    XX.  47 

4th.  From  the  divisions  in  A.  L  and  B.  M,  draw  lines  to  one 
end  of  the  conjugate  axis  at  C;  and,  from  the  divisions  in  B. 
0  and  Jl.  JV,  draw  lines  to  the  other  end  at  D. 

5th.  From  D,  through  the  points  1,  2,  3,  in  A.  S,  draw  lines  to 
intersect  the  lines  1,  2,  3  drawn  from  the  divisions  on  Jl.  L,  and 
in  like  manner  through  B.  S,  to  intersect  the  lines  from  B.  M. 
These  points  of  intersection  are  points  in  the  curve. 

6th.  From  C,  through  the  divisions  1,  2,  3,  on  S.  Jl  and  S.  B, 
cTraw  lines  to  intersect  the  lines  1,  2,  3  drawn  from  Jl.  N  and 
B.  0,  which  will  give  the  points  for  drawing  the  other  half  of 
the  curve. 

7th.    Through  the  points  of  intersection,  trace  the  curve. 

NOTE. — If  required  on  a  small  scale,  the  curve  can  be  drawn  by  hand  ;  but,  if 
required  on  a  large  scale,  for  practical  purposes,  it  is  best  to  drive  sprigs  at 
the  points  of  intersection,  and  bend  a  thin  flexible  strip  of  pine  around  them, 
for  the  purpose  of  tracing  the  curve.  Any  number  of  points  may  be  found 
by  dividing  the  lines  into  the  requisite  number  of  parts. 


FIGURE  4. 


Is  a  semi-ellipsis,  drawn  on  the  conjugate  axis  by  the  same  method, 

in  which  JL.  B  is  the  transverse,  and  C.  D  the  conjugate  axis. 
NOTE. — This  method  will  apply  to  an  ellipsis  of  any  length  or  breadth. 


PROBLEM  51.     FIG.  2. 
To  draw  an  Ellipsis  with  a  Trammel. 

The  TRAMMEL  shewn  in  the  diagram  is  composed  of  two  pieces 
of  wood  halfened  together  at  right  angles  to  each  other,  with  a 
groove  running  through  the  centre  of  each,  the  groove  being 
wider  at  the  bottom  than  at  the  top.  /.  K.  L  is  another  strip  of 
wood  with  a  point  at  /,  or  with  a  hole  for  inserting  a  pencil  at  /, 
and  two  sliding  buttons  at  TTand  L;  the  buttons  are  generally  at- 
tached to  small  morticed  blocks  sliding  over  the  strip,  with  wedges 
or  screws  for  securing  them  in  the  proper  place;  (the  pins  are 
only  shewn  in  the  diagram,)  the  buttons  attached  to  the  pins  are 
made  to  slide  freely  in  the  grooves. 


48  PLATE    XX. 

Mode  of  Setting  the  Trammel. 

1st.  Make  the  distance  /.  K  equal  to  the  semi-conjugate  axis,  and 
the  distance  from  /  to  L  equal  to  the  semi-transverse  axis. 

2nd.  Set  the  grooved  strips  to  coincide  with  the  axes  of  the  ellipsis, 
and  secure  them  there. 

3rd.  Move  the  point  /  around  and  it  will  trace  the  curve  correctly. 

NOTE. — This  is  a  very  useful  instrument,  and  was  formerly  used  very  fre- 
quently by  carpenters  to  lay  off  their  work,  and  also  by  plasterers  to  run  their 
mouldings  around  elliptical  arches,  &c.,  the  mould  occupying  the  position 
of  the  point  I.  It  was  rare  then  to  find  a  carpenter's  shop  without 
a  trammel  or  to  find  a  good  workman  who  was  not  skilled  in  the  use  of  it ; 
but  since  Grecian  architecture  with  its  horizontal  lintels  has  taken  the  place 
of  the  arch,  it  is  seldom  a  trammel  is  required,  and  when  required,  much 
more  rare  to  find  one  to  use ;  but  as  it  is  sometimes  wanted,  and  few  of 
our  young  mechanics  know  how  to  apply  it,  at  the  risk  of  being  thought 
tedious,  we  have  been  thus  minute  in  its  description. 


PROBLEM  52.     FIG.  3. 


To  describe  the  Hyperbola  from  the  Cone. 

1st.  Draw  the  line  Jl.  C.  B  and  make  C.  B  and  C.  A  each  equal 
to  C.  B,  fig.  4;  plate  xix,  then  A.  B  will  be  equal  to  the  base  of 
the  hyperbola. 

2nd.  Perpendicular  to  Jl.  B,  draw  A.  E  and  B.  F,  and  make  them 
equal  to  C.  D,  fig.  4,  plate  xix. 

3rd.  Join  E.  F,  from  C  erect  a  perpendicular  C.  D.  0,  and  make 
C.  0  equal  to  C.  0,  fig.  4,  plate  xix. 

4th.  Divide  JL.  E  and  B.  F  each  into  any  number  of  equal  parts^ 
say  4,  and  divide  B.  C  and  C.  Jl  into  the  same  number,  and 
number  them  as  in  the  diagram. 

5th.  From  the  points  of  division  on  Jl.  E  and  B.  F,  draw  lines  to  D. 

6th.  From  the  points  of  division  in  Jl.  B,  draw  lines  toward  0;  and 
the  points  where  they  intersect  the  other  lines  with  the  same 
numbers  will  be  points  in  the  curve.  The  curve  Jl.  D.  B  is  the 
section  of  the  cone  through  the  line  C.  D,  fig.  4,  plate  xix. 


MJJI'SIS  AND  HYPERBOLA 


49 


PLATE    XXI. 

PARABOLA    AND    ITS    APPLICATION 


PROBLEM  53.     FIG.  1. 
To  describe  the  Parabola  by  Tangents. 

1st.  Draw  jl.  P.  B,  make  A.   P  and  P.  B  each  equal  to  Jl.  P} 

fig.  4;  plate  xix. 
2nd.  From  P  draw  P.    Q.  R  perpendicular  to  Jl.   B,  and  make 

P.  R  equal  to  twice  the  height  of  P.  Q,  fig.  4,  plate  xix. 
3rd.  Draw  Jl.  R  and  B.  R,  and  divide  them  each  into  the  same 

number  of  equal  parts,  say  eight;  number  one  side  from  Jl  to  R, 

and  the  other  side  from  R  to  B. 
4th.  Join  the  points  1.  1. — 2.  2. — 3.  3,  &c.;  the  lines  so  drawn 

will  be  tangents  to  the  curve,  which  should  be  traced  to  touch 

midway  between  the  points  of  intersection. 
The  curve  Jl.  Q.  B  is  a  section  of  the  cone  through  P.  Q,  fig.  4; 

plate  xix. 


PROBLEM  54.     FIG.  2. 
To  describe  the  Parabola  by  another  method. 

Let  Jl.  B  be  the  width  of  the  base  and  P.  Q  the  height  of  the 

curve. 

1st.  Construct  the  parallelogram  Jl.  B.  C.  D. 
2nd.  Divide  Jl.  C  and  Jl.  P—P.  B  and  B.  D  respectively  into  a 

similar  number  of  equal  parts;  number  them  as  in  the  diagram. 
3rd.  From  the  points  of  division  in  Jl.  C  and  B.  D,  draw  lines 

to  Q. 
4th.  From  the  points  of  division  on  A.  B  erect  perpendiculars  to 

intersect  the  other  lines ;  the  points  of  intersection  are  points  in 

the  curve. 


50  PLATE  XXI. 

PROBLEM  55.     FIG.  3. 


To  describe  a  Parabola  by  continued  motion,  with  a  Ruler,  String 

and  Square. 

Let  C.  D  be  the  width  of  the  curve  and  H.  J  the  height. 

1st.  Bisect  H.  D  in  K,  draw  J.  K  and  K.  E  perpendicular  to  J. 

K,  cutting  J.  H  extended  in  E.     Then  take  the  distance  H.  E 

and  set  it  off  from  J  to  F,  then  F  is  the  focus. 
2nd.    At  any  convenient  distance  above  J,  fasten  a  ruler  Jl.  B, 

parallel  to  the  base  of  the  parabola  C.  D. 
3rd.    Place  a  square  S,  with  one  side  against  the  edge  of  the 

ruler,  Jl.  B,  the  edge  0.  JV*  of  the  square  to  coincide  with  the 

line  E.  J. 
4th.  Tie  one  end  of  a  string  to  a  pin  stuck  in  the  focus  at  F,  place 

your  pencil  at  J,  pass  the  string  around  it,  and  bring  it  down  to 

JV*,  the  end  of  the  square,  and  fasten  it  there. 
5th.    With  the  pencil  at  J,  against  the  side  of  the  square,  and  the 

string  kept  tight,  slide  the  square  along  the  edge  of  the  ruler 

towards  Jl;  the  pencil  being  kept  against  the  edge  of  the  square, 

with  the  string  stretched,  will  describe  one  half  of  the  parabola, 

J    C. 
6th.    Turn  the  square  over,  and  draw  the  other  half  in  the  same 

manner. 


DEFINITIONS. 

1st.  The  FOCUS  of  a  parabola  is  the  point  F,  about   which  the 

string  revolves.     The  edge  of  the  ruler  Jl.  B,  is  the  directrix  of 

the  parabola. 
2nd.    The  AXIS  is  the  line  J.  H,  passing  through  the  focus,  and 

perpendicular  to  the  base  C.  D. 
3rd.    The  principal  VERTEX,  is  the  point  J,  where  the  top  of  the 

axis  meets  the  curve. 
4th.    The  PARAMETER,  is  a  line  passing  through  the  focus,  parallel 

to  the  base,  terminated  at  each  end  by  the  curve. 
5th.    Any  line,  parallel  to  the  axis,  and  terminating  in  the  curve, 

is  called  a  DIAMETER,  and  the  point  where  it  meets  the  curve,  is 

called  the  vertex  of  that  diameter. 


PLATE  XXI.  51 

PROBLEM  56.     FIG.  4. 


To  apply  the  Parabola  to  the  construction  of  Gothic  Jlrches. 

1st.  Draw  Ji.  B,  and  make  it  equal  to  the  width  of  the  arch 
at  the  base. 

2nd.  Bisect  Jl.  B  in  E,  draw  E.  F  perpendicular  to  A.  B,  and 
make  E.  F  equal  to  the  height  of  the  arch. 

3rd.    Construct  the  parallelogram  Jl.  B.  C.  D. 

4th.  Divide  E.  F  into  any  number  of  equal  parts,  and  D.  F 
and  F.  C  each  into  a  similar  number,  and  number  them  as  in 
the  diagram. 

5th.  From  the  divisions  on  F.  D,  draw  lines  to  JL,  and  from  the 
divisions  on  F.  C,  draw  lines  to  B. 

6th.  Through  the  divisions  on  E.  F,  draw  lines  parallel  to  the 
base,  to  intersect  the  other  lines  drawn  from  the  same  numbers 
on  Z).  C.  The  points  of  intersection  are  points  in  the  curve 
through  which  it  may  be  traced. 

NOTE. — If  we  suppose  this  diagram  to  be  cut  through  the  line  E.  F,  and 
turned  around  until  E.  A  and  E.  B  coincide,  it  will  form  a  parabola, 
drawn  by  the  same  method  as  fig.  2 ;  and,  if  we  were  to  cut  fig.  2  by  the 
line  P.  Q,  and  turn  it  around  until  P.  A  and  P.  B  coincide,  it  would  form 
a  gothic  arch,  described  by  the  same  method  as  fig.  4 ;  and,  if  the  propor- 
tions of  the  two  figures  were  the  same,  the  curves  would  exactly  coincide. 


PLATE    XXII. 


PROBLEM  57. 


Given  the  position  of  three  points  in  the  circumference  of  a  Cylin- 
der, and  their  respective  heights  from  the  base,  to  find  the  section 
of  the  segment  of  the  Cylinder  through  these  three  points. 

1st.  Let  A.  B.  C  be  three  points  in  the  circumference  of  the  base 
of  the  cylinder,  immediately  under  the  three  given  points,  and 
A'.  /)', —  C.  P,  and  B'. — Ef9 — the  height  of  the  given  points, 
respectively,  above  the  base. 

2nd.  Join  the  points  A  and  B,  and  draw  A.  D, —  C.  F,  and  B. 
E,  perpendicular  to  JL.  B. 


52  PLATE  XXII. 

3rd.  Make  JL.  D  equal  to  J¥ .  D',  the  height  of  the  given  point 
above  the  base  at  A, — make  B.  E  equal  to  B'.  E',  and  C.  F 
equal  to  C'.  F. 

4th.    Produce  B.  Jl  and  E.  D,  to  meet  each  other  in  H. 

5th.    Draw  C.  G  parallel  to  B.  H,  and  F.  G  parallel  to  E.  H. 

6th.    Join   G.  #. 

7th.  In  G.  H,  take  any  point  as  G,  and  draw  G.  K  perpendicu- 
lar to  G.  C,  cutting  .#.  #  in  JT. 

8th.  From  the  point  K,  draw  JT.  /  perpendicular  to  E.  H,  cut- 
ting E.  H  in  Z,. 

9th.  From  H,  with  the  radius  H.  G,  describe  the  arc  G.  /,  cut- 
ting K.  L  in  /,  and  join  H.  I. 

10th.  Divide  the  circumference  of  the  segment  Jl.  C.  B  into 
any  number  of  equal  parts,  and  from  the  points  of  division.,  draw 
lines  to  Jl.  B,  parallel  to  G.  H,  cutting  Jl.  B  in  1,2,  3,  &c. 

llth.  From  the  points  1,  2,  3,  &c.  in  Jl  B,  draw  lines  parallel  to 
B.  E,  cutting  the  line  D.  E  in  1,  2,  3,  &c. 

12th.  From  the  points  1,  2,  3,  &c.  in  D.  E,  draw  lines  parallel 
to  H.  I,  and  make  1.  0  equal  to  1.  1  on  the  base  of  the  cylin- 
der, make  2.  0  equal  to  2.  2,  3.  0  equal  to  3.  3,  &c. 

1 3th.  Through  the  points  0,  0,  0,  &c.,  trace  the  curve,  which 
will  be  the  contour  of  the  section  required. 

NOTE. — It  will  be  perceived  that  the  line  2.  2  intersects  A.  B  in  A,  and  that 
the  line  A.  D  obviates  the  necessity  of  drawing  the  perpendicular  from  2, 
as  required  by  the  1 1th  step  in  the  problem. 


PLATE   XXIII. 

PROBLEM  58.  FIG  1. 
To  draw  the  Boards  for  covering  Circular  Domes. 

To  lay  the  boards  vertically.  Let  Jl.  D.  C  be  half  the  plan 
of  the  dome ;  let  D.  C  represent  one  of  the  ribs,  and  E.  F  the 
width  of  one  of  the  boards. 

1st.    Draw  D.  0,  and  continue  the  line  indefinitely  toward  H. 

2nd.    Divide  the  rib  D.  C  into  any  number  of  equal  parts,  and 


TO  FIND  THE  SECTION 
OF  THE  SEGMENT  OF  A  CYLINDIW 
THR O  UGH  THREE  GIVEN  POINTS. 


TO  DfLlW  THE  BOARDS  FOR  COVERING 
HEMISPHERICAL  DOMES. 


I'},/.  / 


PLATE    XXIII,  53 

from  the  points  of  division,  draw  lines  parallel  to  Jl.  C,  meeting 
D.  0  in  1,  2,  3,  &c. 

3rd.  With  an  opening  of  the  dividers  equal  to  one  of  the  divi- 
sions on  D.  C,  step  off  from  D  toward  H,  the  same  number  of 
parts  as  D.  C  is  divided  into,  making  the  right  line  D.  H}  nearly 
equal  to  the  curve  D.  C. 

4th.    Join  E.    0  and  F.   0. 

5th.  Make  1.  c — 2.  d — 3.  e — 4./  and  5  g,  on  each  side  of  D.  H, 
equal  to  1.  c — 2.  d — 3.  e,  &c.  on  D.  0. 

6th.  Through  the  points  c.  d.  e.  f.  g,  trace  the  curve,  which  will 
be  an  arc  of  a  circle ;  and  if  a  series  of  boards  made  in  the  same 
manner,  be  laid  on  the  dome,  the  edges  will  coincide. 

NOTE. — In  practice,  where  much  accuracy  is  required,  the  rib  shoulc  be  di- 
vided into  at  least  twelve  parts. 


PROBLEM  59.     FIG.  2. 
To  lay  the  boards  Horizontal^ 

Let  Jl.  B.  C  be  the  vertical  section  of  a  dome  through  its  axis. 

1st.    Bisect  Jl.  C  in  D,  and  draw  D.  P  perpendicular  to  A.  C. 

2nd.  Divide  the  arc  Jl.  B  into  such  a  number  of  equal  parts,  that 
each  division  may  be  less  than  the  breadth  of  a  board.  (If  we 
suppose  the  boards  to  be  used  to  be  of  a  given  length,  each  di- 
vision should  be  made  so  that  the  curves  struck  on  the  hollow 
side  should  touch  the  ends,  and  the  curves  on  the  convex  side 
should  touch  the  centre.) 

3rd.  From  the  points  of  division,  draw  lines  parallel  to  Jl.  C  to 
meet  the  opposite  side  of  the  section.  Then  if  we  suppose  the 
curves  intercepted  by  these  lines  to  be  straight  lines,  (and  the 
difference  will  be  small,)  each  space  would  be  the  frustrum  of  a 
cone,  whose  vertex  would  be  in  the  line  D.  P,  and  the  vertex 
of  each  frustrum  would  be  the  centre  from  which  to  describe  the 
curvature  of  the  edges  of  the  board  to  fit  it. 

4th.  From  1  draw  a  line  through  the  point  2,  to  meet  the  line  D 
P  in  E;  then  from  Ey  with  a  radius  equal  to  E.  1,  describe  the 
curve  1.  Z,  which  will  give  the  lower  edge  of  the  board,  and 
with  a  radius  equal  to  E.  2,  describe  the  arc  2.  K,  which  will 
give  the  upper  edge.  The  line  L.  K  drawn  to  E,  will  give  the  cut 
for  the  end  of  a  board  which  will  fit  the  end  of  any  other  board 
cut  to  the  same  angle. 


54  PLATE    XXIII. 

5th.  From  2  draw  a  line  through  3,  meeting  D.  P  in  F.  From 
3,  draw  a  line  through  4,  meeting  D.  P  in  G,  and  pioceed  for 
each  board,  as  in  paragraph  4. 

6th.  If  from  C  we  draw  a  line  through  My  and  continue  it  up- 
ward, it  would  require  to  be  drawn  a  very  great  distance  before 
it  would  meet  D.  P ;  the  centre  would  consequently  be  incon- 
veniently distant. 

For  the  bottom  board,  proceed  as  follows : 

1st.    Join  A.  M,  cutting  D.  P  in  JV,  and  join  N.  1. 

2nd.  Describe  a  curve,  by  the  methods  in  Problems  34  or  35, 
Plate  11,  through  1.  JV*.  M,  which  will  give  the  centre  of  the 
board,from  which  the  width  on  either  side  may  be  traced. 


PLATE   XXIV. 

CONSTRUCTION    OF    ARCHES. 


Arches  in  architecture  are  composed  of  a  number  of  stones  arrang- 
ed symmetrically  over  an  opening  intended  for  a  door,  window, 
&,c.,  for  the  purpose  of  supporting  a  superincumbent  weight.  The 
depth  of  the  stones  are  made  to  vary  to  suit  each  particular  case, 
being  made  deeper  in  proportion  as  the  width  of  an  opening  be- 
comes larger,  or  as  the  load  to  be  supported  is  increased ;  the  si/,e 
of  the  stones  also  depends  much  on  the  quality  of  the  material  of 
which  they  are  composed :  if  formed  of  soft  sandstone  they  will 
require  to  be  much  deeper  than  if  formed  of  granite  or  some  other 
hard  strong  stone. 


DEFINITIONS.     FIG.  2. 

1st.  The  SPAN  of  an  arch  is  the  distance  between  the  points  of 
support,  which  is  generally  the  width  of  the  opening  to  be  cov- 
ered, as  Jl.  B.  These  points  are  called  the  springing  points ;  the 
mass  against  which  the  arch  rests  is  called  the  ABUTMENT. 

3rd.  The  RISE,  HEIGHT  or  VERSED  SINE  of  an  arch,  is  the  dis- 
tance from  C  to  D. 

2nd.  The  SPRINGING  LINE  of  an  arch  is  the  line  Jl.  B,  being  a 
horizontal  line  drawn  across  the  tops  of  the  support  where  the 
arch  commences. 


PLATE    XXIV.  55 

4th.  The  CROWN  of  an  arch  is  the  highest  point,  as  Z>. 

5th.  VOUSSOIRS  is  the  name  given  to  the  stones  forming  the  arch. 

6th.  The  KEYSTONE  is  the  centre  or  uppermost  voussoir  D,  so 
called;  because  it  -is  the  last  stone  set,  and  wedges  or  keys  the 
whole  together.  Keystones  are  frequently  allowed  to  project 
from  the  face  of  the  wall,  and  in  some  buildings  are  very  elabor- 
ately sculptured. 

7th.  The  INTRADOS  or  SOFFIT  of  an  arch  is  the  under  side  of 
the  voussoirs  forming  the  curve. 

8th.  The  EXTRADOS  or  BACK  is  the  upper  side  of  the  voussoirs. 

9th.  The  THRUST  of  an  arch  is  the  tendency  which  all  arches  have 
to  descend  in  the  middle,  and  to  overturn  or  thrust  asunder  the 
points  of  support. 

NOTE. — The  amount  of  the  thrust  of  an  arch  depends  on  the  proportions  be- 
tween the  rise  and  the  span,  that  is  to  say,  the  span  and  weight  to  be  sup- 
ported being  definite;  the  thrust  will  be  diminished  in  proportion  as  the 
rise  of  the  arch  is  increased,  and  the  thrust  will  be  increased  in  proportion 
as  the  crown  of  the  arch  is  lowered. 

10th.  The  JOINTS  of  an  arch  are  the  lines  formed  by  the  adjoining 
faces  of  the  voussoirs;  these  should  generally  radiate  to  some  de- 
finite point,  and  each  should  be  perpendicular  to  a  tangent  to  the 
curve  at  each  joint.  In  all  curves  composed  of  arcs  of  circles,  a 
tangent  to  the  curve  at  any  point  would  be  perpendicular  to  a 
radius  drawn  from  the  centre  of  the  circle  through  that  point, 
consequently  the  joints  in  all  such  arches  should  radiate  to  the 
centre  of  the  circle  of  which  the  curve  forms  a  part. 

llth.  The  BED  of  an  arch  is  the  top  of  the  abutment;  the  shape 
of  the  bed  depends  on  the  quality  of  the  curve,  and  will  be  ex- 
plained in  the  diagrams. 

12th.  A  RAMPANT  ARCH  is  one  in  which  the  springing  points  are 
not  in  the  same  level. 

13th.  A  STRAIGHT  ARCH,  or  as  it  is  more  properly  called,  a  PLAT 
BAND,  is  formed  of  a  row  of  wedge-shaped  stones  of  equal  depth 
placed  in  a  horizontal  line,  the  upper  ends  of  the  stones  being 
broader  than  the  lower,  prevents  them  from  falling  into  the  void 
below. 

14th.  Arches  are  named  from  the  shape  of  the  curve  of  the  under 
side,  and  are  either  simple  or  complex.  I  would  define  simple 
curves  to  be  those  that  are  struck  from  one  centre,  as  any  segment 
of  a  circle,  or  by  continued  motion,  as  the  ellipsis,  parabola,  hy- 
perbola, cycloid  and  epicycloid;  and  COMPLEX  ARCHES  to  be 


56  PLATE    XXIV. 

those  described  from  two  or  more  fixed  centres,  as  many  of  the 
Gothic  or  pointed  arches.  The  simple  curves  have  all  been  de- 
scribed in  our  problems  of  practical  Geometry;  we  shall  however 
repeat  some  of  them  for  the  purpose  of  showing  the  method  of 
drawing  the  joints. 


PROBLEM  60.     FIG.   1. 
To  describe  a  Segment  or  Scheme  Jlrch,  and  to  draw  the  Joints. 

1st.  Let  E  and  F  be  the  abutments,  and  0  the  centre  for  describ- 
ing the  curve. 

2nd.  With  one  foot  of  the  dividers  in  0,  and  the  distance  0.  F, 
describe  the  line  of  the  intrados. 

3rd.  Set  off  the  depth  of  the  voussoirs,  and  with  the  dividers  at 
0,  describe  the  line  of  the  extrados. 

4th.  From  E  and  F  draw  lines  radiating  to  0,  which  gives  the 
line  of  the  beds  of  the  arch.  This  line  is  often  called  by  masons 
a  skew-back. 

5th.  Divide  the  intrados  or  extrados,  into  as  many  parts  as  you 
design  to  have  stones  in  the  arch,  and  radiate  all  the  lines  to  0, 
which  will  give  the  proper  direction  of  the  joints. 

6th.  If  the  point  0  should  be  at  too  great  a  distance  to  strike  the 
curve  conveniently,  it  may  be  struck  by  Problem  34  or  35, 
Plate  1 1  ;  and  the  joints  may  be  found  as  follows  :  Let  it  be  de- 
sired to  draw  a  joint  at  2,  on  the  line  of  the  extrados ;  from  2 
set  off  any  distance  on  either  side,  as  at  1  and  3;  and  from  1  and 
3,  with  any  radius,  draw  two  arcs  intersecting  each  other  at  4 — 
then  from  4  through  2  draw  the  joint  which  will  be  perpendicular 
to  a  tangent,  touching  the  curve  at  2.  This  process  must  be  re- 
peated for  each  joint.  The  keystone  projects  a  little  above  and 
below  the  lines  of  the  arch. 


PROB.  61.     FIG.  2. — THE  SEMICIRCULAR  ARCH. 

This  requires  but  little  explanation.  Ji.  B  is  the  span  and  C  the 
centre,  from  which  the  curves  are  struck,  and  to  which  the  lines 
of  all  the  joints  radiate.  The  centre  C  being  in  the  springing 
line  of  the  arch  the  beds  are  horizontal. 


UFI7EESIT7 

.:' 


Plate  :J/ 


JOISTS  AY  .1/1 C//KS 


/'/,/.    V 


/'///. .y. 


• 


PLATE  XXIV.  57 

PROB.  62.     FIG.  3. — THE  HORSE  SHOE  ARCH 


Is  an  arc  of  a  circle  greater  than  a  semicircle,  the  centre  0  being 
above  the  springing  line. 

This  arch  is  also  called  the  SARACENIC  or  MORESCO  arch,  because 
of  its  frequent  use  in  these  styles  of  architecture.  The  joints 
radiate  to  the  centre,  as  in  fig.  2.  The  joint  at  5,  below  the 
horizontal  line,  also  radiates  to  0.  This  may  do  very  well  for  a 
mere  ornamental  arch,  that  has  no  weight  to  sustain;  but  if,  as  in 
the  diagram,  the  first  stone  rests  on  a  horizontal  bed,  it  would  be 
larger  on  the  inside  than  on  the  outside,  and  would  be  liable  to 
be  forced  out  of  its  position  by  a  slight  pressure,  much  more  so 
than  if  the  joint  were  made  horizontal,  as  at  6.  These  remarks 
will  also  apply  to  fig.  4,  Plate  25. 


PROBLEM  63.     FIG.  4. 
To  describe  an  Ogee  Jlrch,  or  an  Jlrch  of  Contrary  Flexure. 

NOTE. — This  arch  is  seldom  used  over  a  large  opening,  but  occurs  frequently 
in  canopies  and  tracery  in  Gothic  architecture,  the  rib  of  the  arch  being 
moulded. 

1st.    Let  JL.  E  be  the  outside  width  of  the  arch,  and  C.  D  the 

height,  and  let  JL.  E  be  the  breadth  of  the  rib. 
2nd.    Bisect  Jl.  B  in  C,  and  erect  the  perpendicular  C.  D;  bisect 

Jl.  C  in  F,  and  draw  F.  J  parallel  to  C.  D. 
3rd.  Through  D  draw  J".  K  parallel  to  Jl.  B,  and  make  D.  K 

equal  to  D.  J. 
4th.    From  F  set  off  F.  G,  equal  to  Jl.  E  the  breadth  of  the  rib, 

and  make  C.  H  equal  to  C.  G. 
5th.  Join  G.  J  and  H.  K;  then  G  and  H  will  be  the  centres  for 

drawing  the  lower  portion  of  the  arch,  and  J  and  K  will  be  the 

centres  for  describing  the  upper  portion,  and  the  contrary  curves 

will  meet  in  the  lines  G.  J  and  H.  K. 


PROBLEM  64.     FIG.  5. 


To  draw  the  Joints  in  an  Elliptic  Jlrch. 

Let  Jl.  B  be  the  span  of  the  arch,  C.  D  the  rise,  and  F.  F  the 
foci,  from  which  the  line  of  the  intrados  may  be  described. 


58  PLATE  XXIV. 

The  voussoirs  near  the  spring  of  the  arch  are  increased  in  depth, 
as  they  have  to  bear  more  strain  than  those  nearer  the  crown ; 
the  outer  curve  is  also  an  ellipsis,  of  which  Hand  If  are  the  foci. 

To  draw  a  joint  in  any  part  of  the  curve,  say  at  5. 

1st.  From  F  and  F  the  foci,  draw  lines  cutting  each  other  in  the 
given  point  5,  and  continue  them  out  indefinitely. 

2nd.  Bisect  the  angle  5  by  Problem  11,  Page  18,  the  line  of 
bisection  will  be  the  line  of  the  joint. 

The  joints  are  found  at  the  points  1  and  3  in  the  same  manner. 

3d.  If  we  bisect  the  internal  angle,  as  for  the  joints  2  and  4,  the 
result  will  be  the  same. 

4th.  To  draw  the  corresponding  joints  on  the  opposite  side  of 
the  arch,  proceed  as  follows  : 

5th.  Prolong  the  line  C.  D  indefinitely  toward  E,  and  prolong 
the  lines  of  bisection  1,  2,  3,  4  and  5,  to  intersect  C.  E  in  1,  2, 
3,  &c.,  and  from  those  points  draw  the  corresponding  joints  be- 
tween JL  and  D. 


PLATE    XXV. 

TO  DESCRIBE  GOTHIC  ARCHES  AND  TO  DRAW  THE  JOINTS, 


The  most  simple  form  of  Pointed  or  Gothic  arches  are  those  com- 
posed of  two  arcs  of  circles,  whose  centres  are  in  the  springing 
line. 


FIGURE  1. — THE  LANCET  ARCH. 


When  the  length  of  the  span  Jl.  B  is  much  less  than  the  length 
of  the  chord  Jl.  C,  as  in  the  diagram,  the  centres  for  striking  the 
curves  will  be  some  distance  beyond  the  base,  as  shewn  by  the 
rods;  the  joints  all  radiate  to  the  opposite  centres. 

FIG.  2. — THE  EQUILATERAL  ARCH. 

When  the  span  D.  E,  and  the  chords  D.  F  and  E.  F  form  an 
equilateral  triangle,  the  arch  is  said  to  be  equilateral,  and  the 


PLATE    XXV.  59 

centres  are  the  points  D  and  E  in  the  base  of  the  arch,  to  which 
all  the  joints  radiate. 


FIG.  3. — THE  DEPRESSED  ARCH 


Has  its  centres  within  the  base  of  the  arch,  the  chords  being  shorter 
than  the  span ;  the  joints  radiate  to  the  centres  respectively. 

NOTE. — There  are  no  definite  proportions  for  Gothic  arches,  except  for  the 
equilateral ;  they  vary  from  the  most  acute  to  those  whose  centres  nearly 
touch,  and  which  deviate  but  little  from  a  semicircle. 


FIG.  4. — THE  POINTED  HORSESHOE  ARCH. 

This  diagram  requires  no  explanation ;  the  centres  are  above  the 
springing  line.     See  fig.  3;  plate  xxiv;  page  57. 


FIGURE  5. 


To  describe  the  Four   Centred  Pointed  Jlrch. 

1st.  Let  JL.  B  be  the  springing  line,  and  E.  C  the  height  of  the 
arch. 

2nd.  Draw  B.  D  parallel  to  E.  C,  and  make  it  equal  to  two- 
thirds  of  the  height  of  E.  C. 

3rd.  Join  D.  C,  and  from  C  draw  C.  L  perpendicular  to  C.  D. 

4th.  Make  C.  G  and  B.  F  both  equal  to  B.  D. 

5th.  Join  G.  F,  and  bisect  it  in  H,  then  through  H  draw  H.  L 
perpendicular  to  G.  F  meeting  C.  L  in  L. 

6th.  Join  L.  F,  and  continue  the  line  to  N.  Then  L  and  F  are 
the  centres  for  describing  one-half  of  the  arch,  and  the  curves  will 
meet  in  the  line  L.  F.  JY. 

7th.  Draw  L.  M  parallel  to  Jl.  B,  make  0.  M  equal  to  0.  L, 
and  E.  K  equal  to  E.  F.  Then  K  and  M  are  the  centres  for 
completing  the  arch,  and  the  curves  will  meet  in  the  line  M.  K.  P. 

8th.  The  joints  from  P  to  C  will  radiate  to  M;  from  C  to  JVthey 
will  radiate  to  L ;  from  JY  to  B  they  will  radiate  to  F,  and  from 
P  to  Jl  they  will  radiate  to  K. 

NOTE  1. — As  the  joint  at  P  radiates  to  both  the  centres  K  and  Jlf,  and  the 
joint  at  JV"  radiates  both  to  F  and  I/,  the  change  of  direction  of  the  lower 
joints  is  easy  and  pleasing  to  the  eye,  so  much  so  that  we  should  be  uncon- 
scious of  the  change,  if  the  constructive  lines  were  removed. 


60  PLATE    XXV. 

NOTE  2. — When  the  centres  for  striking  the  two  centred  arch  are  in  the 
springing  line,  as  in  diagrams  1,  2  and  3,  the  vertical  side ,  of  the  opening 
joins  the  curve,  without  forming  an  unpleasant  angle,  as  it  would  do  if  the 
vertical  lines  were  continued  up  above  the  line  of  the  centres  ;  it  is  true  that 
examples  of  this  character  may  be  cited  in  Gothic  buildings,  but  its  ungrace- 
ful appearance  should  lead  us  to  avoid  it. 


PLATES   XXVI  AND    XXVII 

DESIGN    FOR    A    COTTAGE. 


Fig.  1.  Is  the  elevation  of  the  south-east  front. 

Fig.  2.  Plan  of  the  ground  floor. 

Fig.  3.  Section  through  the  line  E.  F  on  the  plan  fig.  2,  the 
front  part  of  the  house  supposed  to  be  taken  away. 

Fig.  4.  Plan  of  the  chamber  floor. 

This  simple  design  is  given  for  the  purpose  of  shewing  the  method 
of  drawing  the  plans.,  section,  elevation  and  details  of  a  building ; 
it  is  not  offered  as  a  "  model  cottage/'  although  it  would  make  a 
very  comfortable  residence  for  a  small  family. 

The  PLAN  of  a  building  is  a  horizontal  section;  if  we  suppose  the 
house  cut  off  just  above  the  sills  of  the  windows  of  the  second 
floor  and  the  upper  portion  taken  away,  it  would  expose  to  view 
the  whole  interior  arrangement,  shewing  the  thickness  of  the 
walls,  the  situation  and  thickness  of  the  partitions  and  the  position 
of  doors,  windows,  &c.;  all  these  interior  arrangements  are  in- 
tended to  be  represented  by  fig.  4. 

If  we  perform  the  same  operation  above  the  sills  of  the  first  floor 
windows,  the  arrangements  of  that  floor,  including  the  stairs  and 
piazzas,  would  appear  as  in  fig.  2. 

A  SECTION  of  a  building  is  a  vertical  plan  in  which  the  thicknesses 
of  the  walls,  sections  of  the  fire-places  and  flues,  sizes  and  direc- 
tion of  the  timbers  for  the  floors  and  roof,  depth  of  the  foundations 
and  heights  of  the  stories  are  shewn,  all  drawn  to  a  uniform  scale. 

If  the  front  of  the  building  is  supposed  to  be  removed,  as  in  fig.  3, 
the  whole  of  the  inside  of  the  back  wall  will  be  seen  in  elevation, 
shewing  the  size  and  finish  of  the  doors  and  windows.,  the  height 
of  the  washboards,  and  the  stucco  cornice  in  the  parlor.  In  looking 


PLATES    XXVI    AND    XXVII.  61 

through  the  door  at  K,  the  first  flight  of  stairs  in  the  back  build- 
ing is  seen  in  elevation. 

If  we  suppose  the  spectator  to  be  looking  in  the  opposite  direction, 
the  back  part  of  the  house  removed,  he  would  see  the  inside  of 
the  front  windows  &c.  instead  of  the  back. 

An  ELEVATION  of  a  building  is  a  drawing  of  the  front,  side  or  back, 
in  which  every  part  is  laid  down  to  a  scale,  and  from  which  the 
size  of  every  object  may  be  measured. 

A  PERSPECTIVE  VIEW  of  a  building,  is  a  drawing  representing  it 
as  it  would  appear  to  a  spectator  from  some  definite  point  of  view, 
and  in  which,  all  objects  are  diminished  as  they  recede  from  the 
eye. 

The  PLANS,  SECTIONS  and  ELEVATIONS,  give  the  true  size  and 
arrangements  of  the  building  drawn  to  a  scale,  and  shew  the 
whole  construction. 

The  PERSPECTIVE  VIEW  should  shew  the  building  complete,  in 
connection  with  the  surrounding  objects,  which  would  enable  the 
proprietor  to  judge  of  the  effect  of  his  intended  improvement. 

The  whole  constitutes  the  DESIGN,  which  for  a  country  house  can- 
not be  considered  complete  without  a  perspective  view. 

To  make  a  design  for  a  dwelling  house,  or  other  building,  it  is  ne- 
cessary before  we  commence  the  drawing,  that  we  should  know 
the  site  on  which  it  is  to  be  erected,  and  the  amount  of  accommo- 
dation required. 

In  choosing  a  site  for  a  country  residence  many  subjects  should  be 
taken  into  consideration ;  for  example,  it  should  be  easy  of  access, 
have  a  good  supply  of  pure  water,  be  on  elevated  ground  to  allow 
the  rain  and  other  water  to  flow  freely  from  it,  but  not  so  high  as 
to  be  exposed  to  the  full  blasts  of  the  winter  storms;  it  should 
have  a  good  prospect  of  the  surrounding  country,  and  above  all, 
it  should  be  in  a  salubrious  locality,  free  from  the  malaria  arising 
from  the  vicinity  of  low  or  marshy  grounds,  with  free  scope  to 
allow  the  house  to  front  toward  the  most  eligible  point  of  the 
compass. 

The  ASPECT  of  a  country  house  is  of  much  importance;  for  if  the 
site  commands  an  extensive  view,  or  pleasant  prospect  in  any  di- 
rection, the  windows  of  the  sitting  and  principal  sleeping  rooms, 
should  front  in  that  direction :  provided  it  does  not  also  face  the 
point  from  which  blow  the  prevailing  storms  of  the  climate,  this 
should  be  particularly  considered  in  choosing  the  site.  Rooms  to 
be  cheerful  and  pleasant,  should  front  south  of  due  east  or  west;  at 


62  PLATES  XXVI  AND  XXVII. 

the  same  time  it  is  desirable  that  the  view  of  disagreeable  or  un- 
sightly objects  should  be  excluded,  and  as  many  as  possible  of  the 
agreeable  and  beautiful  objects  of  the  neighborhood  brought  into 
view.  The  design  before  us,  is  made  to  front  the  south-east,  all 
the  openings  except  two  are  excluded  from  the  north  easterly 
storms,  which  are  the  most  disagreeable  in  the  Atlantic  States; 
the  sun  at  noon  would  be  opposite  the  angle  Jl,  and  would  shine 
equally  on  the  front  and  side,  consequently  the  front  would  have 
the  sun  until  the  middle  of  the  afternoon,  and  the  side  of  the  front 
house  and  front  of  the  back  building  would  have  the  evening  sun, 
rendering  the  whole  dry  and  pleasant. 

The  end  of  the  back  building,  containing  the  kitchen  and  stairs,  is 
placed  against  the  middle  of  the  back  wall  of  the  front  building, 
to  allow  the  back  windows  in  the  parlor,  &c.,  to  be  placed  in  the 
middle  of  each  room.  These  windows  may  be  closed  in  stormy 
weather  with  substantial  shutters;  but  in  warm  weather  they  will 
add  much  to  the  coolness  of  the  rooms,  by  allowing  a  thorough 
ventilation. 

The  broad  projecting  cornice  of  the  house,  and  the  continuous 
piazza,  are  the  most  important  features  in  the  elevation;  besides 
the  advantage  of  keeping  the  walls  dry,  and  throwing  the  rain- 
water away  from  the  foundation,  they  give  an  air  of  comfort, 
which  would  be  entirely  wanting  without  them ;  for  if  we  were 
to  take  away  the  piazza,  and  reduce  the  eave  cornice  to  a  slight 
projection,  the  appearance  would  be  bald  and  meagre  in  the 
extreme. 

The  projection  of  the  piazza  is  increased  on  the  front  and  rear,  to 
give  more  room  to  the  entrances. 

The  front  building  is  36  feet  wide  from  Jl  to  £,  fig.  2,  and  20  feet 
deep  from  Jl  to  D.  The  back  building  is  16  feet  wide,  by  20 
feet  deep.  The  scale  at  the  bottom  of  each  plate  must  be  used 
to  get  the  sizes  of  all  the  minor  parts.  The  height  of  the  first 
story  is  10  feet  in  the  clear,  and  of  the  second  story  8'  6";  these 
heights  are  laid  off  on  a  rod  /?,  to  the  right  of  fig.  1 ;  so  are  also 
the  heights  of  the  windows,  which  shews  at  a  glance,  their  posi- 
tion with  regard  to  the  floors  and  ceilings.  This  method  should 
always  be  resorted  to  in  drawing  an  elevation,  as  it  will  the  better 
enable  the  draughtsman  to  make  room  for  the  interior  finish  of 
the  windows  and  for  the  cornice  of  the  room. 

In  laying  down  a  plan,  the  whole  of  the  outer  walls  should  be  first 
drawn,  and  in  setting  off  openings  and  party  walls,  the  measure- 


PLATES    XXVI    AND    XXVII.  63 

ments  should  be  taken  from  both  corners,  to  prove  tha  you  are 
correct.  For  example,  in  setting  off  the  front  door,  take  the 
width  5.  0  from  the  whole  width  of  the  front,  which  will  leave 
31.  0;  then  lay  off  15'.  6"  from  Jl,  and  also  from  B,  then  if  the 
width  of  5.  0  is  left  between  the  points  so  measured,  you  are 
sure  the  front  door  is  laid  off  correctly ;  as  the  windows  C  and  H 
are  midway  between  the  front  door  and  the  corner  of  the  build- 
ing, the  same  plan  should  be  followed,  and  as  a  general  rule 
that  will  save  trouble  by  preventing  errors,  you  should  never  de- 
pend on  the  measurements  from  one  end  or  corner,  if  you  have 
the  means  of  proving  them  by  measuring  from  the  opposite  end 
also. 

The  winding  steps  in  the  stairs  may  be  dispensed  with  by  adding 
3  steps  to  the  bottom  flight  bringing  it  out  to  the  kitchen  door, 
and  by  adding  1  step  to  the  top  flight ;  or  a  still  better  arrange- 
ment might  be  made  by  adding  3  steps  to  the  bottom  flight,  and 
retaining  two  of  the  winders:  this  would  give  17  risers  instead  of 
16,  the  present  number,  which  would  reduce  the  height  of  each 
to  7  3-4  inches. 

To  ascertain  the  number  of  steps  required  to  a  story,  proceed  as 
follows :  Add  to  the  clear  height  of  the  story  the  breadth  of  the 
joists  and  floor,  which  will  give  the  full  height  from  'the  top  of 
one  floor  to  the  top  of  the  next.  In  constructing  the  stairs  this 
height  is  laid  off  on  a  rod,  and  then  divided  into  the  requisite 
number  of  risers;  but  in  drawing  the  plan,  as  in  the  case  before 
us,  set  down  the  height  in  feet,  inches  and  parts,  and  divide  by 
the  height  you  propose  for  your  rise:  this  will  give  you  the 
number  of  risers.  If  there  is  any  remainder,  it  may  be  divided 
and  added  to  your  proposed  rise,  or  another  step  may  be  added, 
and  the  height  of  the  rise  reduced ;  or  the  height  of  the  story 
may  be  divided  by  the  number  of  risers,  which  will  give  the  exact 
height  of  the  riser  in  inches  and  parts.  For  example  : 

The  clear  height  of  the  story  in  the  design  is    10'.  0."  > ,  ,x  ~,, 

The  breadth  of  the  joist  and  thickness  of  the  floor  1.0.  > 
this  multiplied  by  12  would  give  132  inches,  and  132  inches  di- 
vided by  16,  the  number  of  risers  on  fig.  1,  will  give  81-4  inches; 
or  divided  by  17  would  give  7  3-4  inches  and  a  fraction.  As  the 
floor  of  the  upper  story  forms  one  step,  there  will  be  always  one 
tread  less  than  there  are  risers.  The  vertical  front  of  each  step 
is  called  the  rise  or  riser,  and  the  horizontal  part  is  called  the 
tread  or  step.  When  the  eaves  of  the  house  are  continued 


64  PLATES    XXVI    AND    XXVII. 

around  the  building  in  the  same  horizontal  line  as  in  this  design, 
the  roof  is  said  to  be  hipped,  and  the  rafter  running  from  the  cor- 
ner of  the  roof  diagonally  to  the  ridge  is  called  the  hip  rafter. 


REFERENCES    TO    THE    DRAWINGS. 

Similar  letters  in  the  plans  and  sections  refer  to  the  same  parts :  thus 
T  the  fire-place  of  the  parlor  in  fig.  2,  is  shewn  in  section  at  T, 
fig.  3,  and  M  the  plan  of  the  back  parlor  window  in  fig.  2,  is 
shewn  in  elevation  at  M,  fig.  3. 

Ji.  By  fig.  2,  is  the  plan,  and  Jl.  By  fig.  1,  the  elevation  of  the 
front  wall. 

E.  Fy  fig.  2,  the  line  of  the  section. 

Gy  fig.  2,  the  front  door. 

Ky  fig.  2,  the  plan,  and  K,  fig.  3,  the  elevation  of  the  door  leading 
to  the  back  building. 

Ly  fig.  2,  the  plan,  and  L,  fig.  3,  the  elevation  of  the  first  flight  of 
stairs. 

M  and  N,  fig.  2,  the  plans,  and  M  and  JV,  fig.  3,  elevation  of  the 
back  first  floor  windows. 

0  and  Py  fig.  4,  the  plans,  and  0  and  P,  fig.  3,  elevation  of  cham- 
ber windows. 

Q,  fig.  2,  the  plan,  and  Q,  fig.  3,  section  of  the  parlor  side  window. 

Ry  fig.  4,  the  plan,  and  R,  fig.  3,  section  of  chamber  side  window. 

*S>,  fig.  4,  the  plan,  and  S,  fig.  3,  elevation  of  railing  on  the  landing. 

T,  fig.  2,  the  plan,  and  !F,  fig.  3,  section  of  parlor  fire-place. 

U,  fig.  2,  the  plan,  and  U,  fig.  3,  section  of  breakfast  room  fire-place. 

V  and  Wy  fig.  4,  the  plans,  and  V  and  W9  fig.  3,  sections  of  cham- 
ber fire-place. 

X  and  Yy  fig.  2,  the  plans,  and  X  and  F,  fig.  3,  elevations  of  side 
posts  on  piazza. 

Z.  Z.  Zy  fig.  4,  plans  of  closets. 

a.  a.  a.y  fig.  4,  flues  from  fire-places  of  ground  floor. 

b.  by  fig.  3,  section  of  eave  cornice. 

c.  Cy  fig.  3,  rafters  of  building. 

d.  dy  fig.  3,  rafters  of  piazza. 

e.  e.  e.  e.  e.  e.  e.  e.  e}  joists  of  the  different  stories;  the  ends  of  the 
short  joists  framed  around  the  fire-places  and  flues  are  shewn  in 
dark  sections;   the  projection  around  the  outside  walls  of  fig.  4, 
shews  the  roof  of  the  piazza. 


65 


PLATE    XXVIII. 


DETAILS. 

Fig.  1  is  an  elevation  of  one  pair  of  rafters,  shewing  also  a  section 

through  the  cornice  and  top  of  the  wall. 
JL,  section  of  the  top  of  the  wall. 

B,  ceiling  joist,  the  outside  end  notched  to  receive  the  cornice. 

C,  collar  beam.     D.  D,  rafters. 
E,  raising  plate.    F,  wall  plate. 

G,  cantilever  and  section  of  cornice. 


FIGURE  2. 


Plan  of  First  Floor  Joists. 

Jl)  foundation  of  kitchen  chimney. 

B,  foundation  of  parlor  chimney ;  C,  of  breakfast  room  do. 

D,  double  joist  to  receive  the  partition  dividing  the  stairway  from 
the  kitchen. 

E.  E,  &LC.  double  joists  resting  on  the  walls  and  supporting  the 
short  joists  F.  F.  F,  forming  the  framing  around  the  fire-places. 

The  joists  E.  E.  E  and  D,  are  called  trimming  joists. 

The  short  joists  F.  F.  F  are  called  trimmers,  and  the  joists  G.  G. 

G,  framed  into  the  trimmers  with  one  end  resting  on  the  wall  are 

called  tail  joists. 


PLATE    XXIX. 

DETAILS. 


Fig.  1?  horizontal  section  through  the  parlor  window, 
Jl)  is  the  outside  of  the  wall.    B  the  inside  of  wall. 


66  PLATE    XXIX. 

C,  the  hanging  stile  of  sash  frame. 

D,  the  inside  lining.  E  the  outside  lining 
F,  the  back  lining.      G.  G  the  weights. 
H}  the  stile*  of  the  outside  or  top  sash. 

I,  the  stile  of  the  inside  or  bottom  sash. 

Ky  inside  stop  bead.     L,  jamb  lining. 

My  ground. f    JV,  plastering. 

0,  architrave.    P,  (dotted  line)  the  projection  of  the  plinth. 

*  The  stiles  of  a   sash,  door,  or  any  other  piece  of  framing,  are  the  vertical 

outside  pieces;  the  horizontal  pieces  are  called  rails, 
f  Grounds  are  strips  of  wood  nailed  against  the  wall  to  regulate  the  thickness 

of  the  plastering,  and  to  receive  the  casings  or  plinth. 


FIGURE  2. 


Vertical  Section  through  the  Sills. 

Jl,  outside  of  the  wall. 

Q,  stone  sill  of  the  window. 

R,  wooden  sub-sill. 

S,  bottom  rail  of  sash.     T,  bondtimber 

Uy  framing  under  window,  called  the  back. 

V,  cap  of  the  back.    K,  the  inside  stop  bead. 


FIGURE  3. 
Plinth    of   Parlor. 

M.  My  grounds.    JV",  plastering. 
y  plinth  or  washboard.      W,  the  base  moulding. 

X,  the  floor. 

Many  more  detail  drawings  might  be  made  of  this  design,  and 
where  a  contract  is  to  be  entered  into,  many  more  should  be  made. 
Enough  is  here  given  to  explain  the  method  of  drawing  them; 
their  use  is  to  shew  the  construction  of  each  part,  and  when 
drawn  to  a  large  scale,  as  in  plate  xxix,  a  workman  of  any  in- 
telligence would  be  able  to  get  out  any  part  of  the  work  required. 


Mate  30 
XAI.   /'/.IX  AXD  KUWATIOX 


Plate  31 


CIRCULAR  FLAN  AND  ELEVATION 


67 


PLATE    XXX. 

OCTAGONAL    PLAN    AND    ELEVATION. 

FIG.   i. — HALF  THE  PLAN.     FIG.  2. — ELEVATION. 

This  plate  requires  but  little  explanation,  as  the  dotted  lines  from 

the  different  points  of  the  plan,  perfectly  elucidate  the  mode  of 

drawing  the  elevation. 
The  dotted  line  *#,  shews  the  direction  of  the  rays  of  light  by 

which  the  shadows  are  projected  ;  the  mode  of  their  projection 

will  be  explained  in  Plates  55  and  56. 


PLATE   XXXI. 

CIRCULAR    PLAN    AND    ELEVATION 


This  plate  shews  the  mode  of  putting  circular  objects  in  elevation. 
The  dotted  lines  from  the  different  points  of  the  plan,  determine 
the  widths  of  the  jambs  (sides)  of  the  door  and  windows,  and 
the  projections  of  the  sills  and  cornice.  One  window  is  farther 
from  the  door  than  the  other,  for  the  purpose  of  shewing  the 
different  apparent  widths  of  openings,  as  they  are  more  or  less 
inclined  from  the  front  of  the  picture. 

This,  as  well  as  Plate  30,  should  be  drawn  to  a  much  larger  size 
by  the  learner ;  he  should  also  vary  the  position  and  width  of  the 
openings.  As  these  designs  are  not  intended  for  a  particular 
purpose,  any  scale  of  equal  parts  may  be  used  in  drawing  them. 


68 


PLATE  XXXIL 

ROMAN     MOULDINGS 


Roman  mouldings  are  composed  of  straight  lines  and  arcs  of 
circles. 

NOTE. — Each  separate  part  of  a  moulding,  and  each  moulding  in  an  assem- 
blage of  mouldings,  is  called  a  member. 


FIG.   1. — A  FILLET,  BAND  OR  LISTEL 


Is  a  raised  square  member,  with  its  face  parallel  to  the  surface  on 
which  it  is  placed. 


FIG.  2. — BEAD. 

A  moulding  whose  surface  is  a  semicircle  struck  from  the  centre  K. 

FIG.   3.— TORUS. 

Composed  of  a  semicircle  and  a  fillet.  The  projection  of  the 
circle  beyond  the  fillet,  is  equal  to  the  radius  of  the  circle,  which 
is  shewn  by  the  dotted  line  passing  through  the  centre  L.  The 
curved  dotted  line  above  the  fillet,  and  the  square  dotted  line  be- 
low the  circle,  shew  the  position  of  those  members  when  used 
as  the  base  of  a  Doric  column. 


FIG.  4. — THE  SCOTIA 

Is  composed  of  two  quadrants  of  circles  between  two  fillets.  B 
is  the  centre  for  describing  the  large  quadrant;  JL  the  centre  for 
describing  the  small  quadrant.  The  upper  portion  may  be  made 
larger  or  smaller  than  in  the  diagram,  but  the  centre  JL  must 
always  be  in  the  line  B.  Jl.  The  scotia  is  rarely,  if  ever  used 
alone ;  but  it  forms  an  important  member  in  the  bases  of  columns 


rid  n-  32, 


fiOMAN  MOULDINGS. 


ft. 


rta.  8.          Cvma  fteyersa 


.\  cud tt 


fes"S 


PLATE  XXXII,  69 

FIG,  5. — THE  OVOLO 


Is  composed  of  a  quadrant  between  two  fillets.  C  is  the  centre  for 
describing  the  quadrant.  The  upper  fillet  projects  beyond  the 
curve,  and  by  its  broad  shadow  adds  much  to  the  effect  of 
the  moulding.  The  ovolo  is  generally  used  as  a  bed  moulding, 
or  in  some  other  position  where  it  supports  another  member. 


FIG  6. — THE  CAVETTO, 


Like  the  ovolo,  is  composed  of  a  quadrant  and  two  fillets.  The 
concave  quadrant  is  used  for  the  cavetto  described  from  D  ;  it 
is  consequently  the  reverse  of  the  ovolo.  The  cavetto  is  frequently 
used  in  connection  with  the  ovolo,  from  which  it  is  separated  by 
a  fillet.  It  is  also  used  sometimes  as  a  crown  moulding  of  a  cor- 
nice; the  crown  moulding  is  the  uppermost  member. 


FIG.  7. — THE  CYMA  RECTA 

Is  composed  of  two  arcs  of  circles  forming  a  waved  line,  and 
two  fillets. 

To  describe  the  cyma,  let  I  be  the  upper  fillet  and  JV*  the  lower 
fillet. 

1st.    Bisect  7.  JV*.  in  M. 

2nd.  With  the  radius  M.  N  or  M.  /,  and  the  foot  of  the  divi- 
ders in  JV*  and  M,  successively  describe  two  arcs  cutting  each 
other  in  F,  and  from  M  and  /  with  the  same  radius,  describe 
two  arcs,  cutting  each  other  in  E. 

3rd.  With  the  same  radius  from  E  and  F,  describe  two  arcs 
meeting  each  other  in  M. 

The  proportions  of  this  moulding  may  be  varied  at  pleasure,  by 
varying  the  projection  of  the  upper  fillet. 


FIG.  8. — THE  CYMA  RE  VERSA,  TALON  OR  OGEE. 

Like  the  cyma  recta,  it  is  composed  of  two  circular  arcs  and  two 
fillets;  the  upper  fillet  projects  beyond  the  curve,  and  the  lower 
fillet  recedes  within  it. 

The  curves  are  described  from  G  and  H. 

The  CYMA,  or  CYMA  RECTA  has  the  concave  curve  uppermost. 


70  PLATE    XXXII. 

The  CYMA  REVERSA  has  the  concave  curve  below. 

The  CYMA  RECTA  is  used  as  the  upper  member  of  an  assemblage 

of  mouldings,  for  which  it  is  well  fitted  from  its  light  appearance. 
The  CYMA  REVERSA  from  its  strong  form,  is  like  the  ovolo,  used  to 

sustain  other  members. 
The  dotted   lines  drawn  at  an  angle  of   45°   to  each  moulding, 

shew  the  direction  of  the  rays  of  light,  from  which  the  shadows 

are  projected. 
NOTE. — When  the  surface  of  a  moulding  is  carved  or  sculptured,  it  is  said 

to  be  ENRICHED. 


PLATE     XXXIII. 


GRECIAN    MOULDINGS 

Are  composed  of  some  of  the  curves  formed  by  the  sections  of  a 
cone,  and  are  said  to  be  elliptic,  parabolic,  or  hyperbolic,  taking 
their  names  from  the  curves  of  which  they  are  formed. 


FIGURES  1  AND  2 


To  draw  the  Grecian  ECHINUS  or  OVOLO,  the  fillets  A  and  B,  the 
tangent  C.  B,  and  the  point  of  greatest  projection  at  D  being 
given. 

1st,  Draw  B.  H,  a  continuation  of  the  upper  edge  of  the  under 

fillet. 
2nd.  Through  D,  draw  D.  H  perpendicular  to  B.  H,  cutting  the 

tangent  B.  C  in  C. 
3rd.  Through  £,  draw  B.  G   parallel  to  D.  H,  and  through  D, 

draw  D.  E  parallel  to  H.  B,  cutting  G.  B  in  E. 
4th.  Make  E.  G  equal  to  E.   B,  and  E.  F  equal  to  H.  C,  join 

D.  F. 
5th.  Divide  the  lines  D.  Fand  D.  C  each  into  the  same  number  of 

equal  parts 
6th.  From  the  point  B,  draw  lines  to  the  divisions  1,  2,  3,  &,c. 

in  D.  C. 


PLATE    XXXIII.  71 

7th.  From  the  point  G>  draw  lines  through  the  divisions  in  'D.  F, 

to  intersect  the  lines  drawn  from  B. 
8th.  Through  the  points  of  intersection  trace  the  curve. 
NOTE. — A  great  variety  of  form  may  be  given  to  the  echinus,  by  varying  the 

projections   and  the  inclination  of  the  tangent  B.  C. 
NOTE  2. — If  H.  C  is  less  than  C.  D,  as  in  fig.  1,  the  curve  will  be  elliptic; 

if  H.  C  and  C.  D  are  equal,  as  in  fig.  2,  the  curve  is  parabolic  ;  if  H.  C  be 

made  greater  than  D.  C,  the  curve  will  be  hyperbolic. 
NOTE  3. — The  echinus,  when   enriched  with  carving,  is   generally  cut  into 

figures  resembling  eggs,  with  a  dart  or  tongue  between  them. 


FIGS.  3  AND  4. — THE  GRECIAN  CYMA. 


To  describe  the  Cyma  Recta,  the  perpendicular  height  B.  D  and 
the  projection  A.  D  being  given. 

1st.  Draw  Jl.  C  and  B.  D  perpendicular  to  JL.  D  and  C.  B  par- 
allel to  Jl.  D. 

2nd.  Bisect  Jl.  D  in  E,  and  Jl.  C  in  G  ;  draw  E.  F  and  G.  0, 
which  will  divide  the  rectangle  A.  C.  B.  D  into  four  equal 
rec  tangles. 

3rd.  Make  G.  P  and  0.  K  each  equal  to  O.  H. 

4th.  Divide  Jl.  G — O.  B — Jl.  E  and  B.  F  into  a  similar  number 
of  equal  parts. 

5th.  From  the  divisions  in  Jl.  E  and  F.  B,  draw  lines  to  H ; 
from  P  draw  lines  through  the  divisions  on  Jl.  G  to  intersect  the 
lines  drawn  from  Jl.  E,  and  from  K  through  the  divisions  in  0. 
By  draw  lines  to  intersect  the  lines  drawn  from  F.  B. 

6th.  Through  the  points  of  intersection  draw  the  curve. 

NOTE. — The  curve  is  formed  of  two  equal  converse  arcs  of  an  ellipsis,  of 
which  E.  F  is  the  transverse  axis,  and  P.  H  or  H.  K  the  conjugate.  The 
points  in  the  curve  are  found  in  the  same  manner  as  in  fig.  1,  plate  20. 


FIG.  5. — THE  GRECIAN  CYMA  REVERSA,  TALON  OR  OGEE. 

To  draw  the  Cyma  Reversa,  the  fillet  A,  the  point  C,  the  end  of 
the  curve  B,  and  the  line  B.  D  being  given. 

1st.  From  C,  draw  C.  D,  and  from  B,  draw  B.  E  perpendicular 
to  B.  D,  then  draw  C.  E  parallel  to  B.  D,  which  completes  the 
rectangle. 


72  PLATE    XXXIII. 

2nd.  Divide  the  rectangle  B.  E.  C.  D  into  four  equal  parts,  by 
drawing  F.  G  and  0.  P. 

3rd.  Find  the  points  in  the  curve  as  in  figs.  3  and  4. 

NOTE  1. — If  we  turn  the  figure  over  so  as  to  bring  the  line  F.  G  vertical,  G 
being  at  the  top,  the  point  B  of  fig.  5,  to  coincide  with  the  point  Jl  of  fig.  3, 
it  will  be  perceived  that  the  curves  are  similar,  jP.  G.  being  the  transverse 
axis,  and  JV*.  H  or  M.  H  the  conjugate  axis  of  the  ellipsis. 

NOTE  2. — The  nearer  the  line  B.  D  approaches  to  a  horizontal  position,  the 
greater  will  be  the  degree  of  curvature,  the  conjugate  axis  of  the  ellipsis 
will  be  lengthened,  and  the  curve  become  more  like  the  Roman  ogee. 

FIGURE  6. — THE  GRECIAN  SCOTIA. 


To  describe  the   Grecian  Scotia,  the  position  of  the  fillets  A  and 

B  being  given. 

1st.  Join  Jl.  .#,  bisect  it  in  C,  and  through  C  draw  D.  .E  parallel 

to  B.   G. 
2nd.  Make  C.  D  and  C.  E  each  equal  to  the  depth  intended  to  be 

given  to  the  scotia;    then  JL.  B  will  be  a  diameter  of  an  ellipsis, 

and  D.  E  its  conjugate. 

3rd.  Through  E,  draw  F.  G  parallel  to  A.  B. 
4th.  Divide  Jl.  F  and  B.  G  into  the  same  number  of  equal  parts, 

and  from  the  points  of  division  draw  lines  to  E. 
5th.  Divide  Jl.  C  and  B.  C  into  the  same  number  of  equal  parts, 

as  Jl.  F,  then  from  D  through  the  points  of  division  in  Jl.  B,  draw 

lines  to  intersect  the  others,  which  will  give  points  in  the  curve. 


PLATE   XXXIV. 

PLAN,  SECTION  AND  ELEVATION  OF  A  WHEEL  AND  PINION. 


The  cross  lines  on  Q.  R,  fig.  2,  shewing  the  teeth  of  the  wheel 
and  pinion,  are  drawn  from  the  elevation  as  described  in  Plate  31, 
which  explains  the  method  of  drawing  an  elevation  from  a  circu- 
lar plan. 

This  plate  is  introduced  to  give  the  learner  an  example  for  draw- 
ing machinery ;  it  requires  but  little  explanation,  as  the  relative 


Plate  34 


ELEVATION 


: 


Plat*  35. 


TEETH  OF  WHEELS 


PLATE  XXXIV.  73 

parts  are  plain  and  simple;  the  same  letters  refer  to  the  same 

parts  in  each  figure. 

Thus  JL,  fig.  1,  is  the  end  of  the  axle  of  the  wheel. 
Jl.  By  fig.  2,  the  top  of  the  axle  of  the  wheel. 
Jl.  By  fig.  3,  section  through  the  centre  of  the  wheel. 
C.  Dy  the  axle  of  the  pinion. 

E.  Fy  flanges  of  the  barrel,  with  the  rope  coiled  between  them. 
G.  Hy  bottom  piece  of  frame. 
/.  K.  K.  N,  inclined  uprights  of  frame. 
Ly  top  of  frame.     M.  My  top  cross  pieces  of  frame. 
0.  Py  bearings  of  the  wheel. 
Q.  Ry  plan  and  elevation  of  wheel. 
Ry  intersection  of  wheel  and  pinion. 
S.  Sy  bottom  cross  pieces  of  frame. 
When  two  wheels  engage  each  other,  the  smallest  is  called  a  pinion. 


PLATE    XXXV. 

TO    DRAW    THE    TEETH    OF    WHEELS 


1st.  The  LINE  of  CENTRES  is  the  line  Jl.  B.  D,  fig.  1,  passing 
through  Jif  and  C,  the  centres  of  the  wheel  and  pinion. 

2nd.  The  PROPORTIONAL  or  primitive  diameter  of  the  wheel,  is 
the  line  Jl.  B;  the  proportional  radius  Jl.  K  or  K.  B.  The  true 
radii  are  the  distances  from  the  centres  to  the  extremities  of  the 
teeth. 

3rd.  The  PROPORTIONAL  DIAMETER  of  the  pinion  is  the  line  B. 
D;  the  proportional  radius  C.  B. 

4th.  The  PROPORTIONAL  CIRCLES  or  PITCH  LINES  are  circles  de- 
scribed with  the  proportional  radii  touching  each  other  in  B. 

5th.  The  PITCH  of  a  wheel  is  the  distance  on  the  pitch  circle  in- 
cluding a  tooth  and  a  space,  as  E.  F  or  G.  H}  or  0.  Z),  fig.  2. 

6th.  The  DEPTH  of  a  tooth  is  the  distance  from  the  pitch  circle  to 
the  bottom,  as  L.  K,  fig.  1,  and  the  height  of  a  tooth  is  the  dis- 
tance from  the  pitch  circle  to  the  top  of  the  tooth,  as  L.  M. 


10 


74  PLATE    XXXV. 


To  draw  the  Pitch  Line  of  a  Pinion  to  contain  a  definite  number 
of  Teeth  of  the  same  size  as  in  the  given  wheel  K. 


1st.  Divide  the  proportional  diameter  Ji.  B  of  the  given  wheel 

into  as  many  equal  parts  as  the  wheel  has  teeth,  viz.  16. 
2nd.  With  a  distance  equal  to  one  of  these  parts,  step  off  on  the 

line  B.  D  as  many  steps  as  the  pinion  is  to  contain  teeth,  which 

will  give  the  proportional  diameter  of  the  pinion;  the  diagram 

contains  8. 
3rd.  Draw  the  pitch  circle,  and  on  it  with  the  distance  E.  F,  the 

pitch,  lay  off  the  teeth. 
4th.  Sub-divide  the  pitch  for  the  tooth  and  space,  draw  the  sides 

of  the  teeth  below  the  pitch  line  toward  the  centre,  and  on  the 

tops  of  the  teeth  describe  epicycloids. 

NOTE. — The  circumferences  of  circles  are  directly  as  their  diameters;  if  the 
diameter  of  one  circle  be  four  times  greater  than  another,  the  circumference 
will  also  be  four  times  greater. 

Fig.  2  is  another  method  for  drawing  the  teeth;  Jl.  B  is  the  pitch 
circle  on  which  the  width  of  the  teeth  and  spaces  must  be  laid 
down.  Then  with  a  radius  D.  E  or  D.  Fy  equal  to  a  pitch  and 
a  fourth,  from  the  middle  of  each  tooth  on  the  pitch  circle  as  at  D, 
describe  the  tops  of  the  teeth  E  and  jF,  from  0  describe  the  tops 
of  the  teeth  G  and  D,  and  so  on  for  the  others.  The  sides  of 
the  teeth  within  the  pitch  circle  may  be  drawn  toward  the  centre, 
as  at  F  and  H,  or  from  the  centre  0,  with  a  radius  equal  to  0. 
Q  or  0.  P>  describe  the  lower  part  of  the  teeth  G  and  D. 


CYLINDER  OF  A  LOCOMOTIVE 

Scale  '/8  *  of  full . vi/.r 


Srri/'/)ii  through  *l.H.  !'«/./. 


CYLINDER  OF  A  LOCOMOTIVE 

Scate  '//"'of  lull  size 


.  3.  End  view 


/''/(/.  /.S redo 1 1  //ifau f//i  //.'///'///./  /' 


75 


PLATES  XXXVI  AND   XXXVII. 

PLAN,  SECTIONS  AND  END  ELEVATION  OF  A  CYLINDER  FOR 
A  LOCOMOTIVE  ENGINE. 


Fig.  1.  Top  view  or  Plan. 

Fig.  2.  Longitudinal  Section  through  Jl.  £,  fig.  1. 

Fig.  3.  Elevation  of  the  end  B,  fig.  1. 

Fig.  4.  Transverse  Section  through  G.  H,  fig.  1. 


REFERENCES 

Jl.  Stuffing  box. 

Jl.  B.      Line  of  longitudinal  section. 

C.  Steam  exhaust  port,  or  Exhaust. 

D.  D.      Steam  ports  or  Side  openings. 

E.  Piston  rod. 

F.  Piston  shewn  in  elevation. 

G.  H.      Line  of  transverse  section. 
H.  Exhaust  pipe. 

K.  Packing. 

L-  Gland  or  Follower. 

M.  M.     Heads  of  cylinder. 

JV.  Valve  face. 

The  piston  is  represented  in  the  drawing  as  descending  to  the  bot- 
tom of  the  cylinder;  the  bent  arrows  from  D  to  C,  fig.  2,  and 
from  C  to  H,  fig.  4;  shew  the  course  of  the  steam  escaping  from 
the  cylinder  through  the  steam  port  and  exhaust  port  to  the  ex- 
haust pipe ;  the  other  arrow  at  Z>;  fig.  2;  the  direction  of  the 
steam  entering  the  cylinder. 


76 


PLATE  XXXVIII. 

ISOMETRICAL    DRAWING. 


FIGURE  1. 
To  draw  the  Isometrical  Cube. 

Let  A  be  the  centre  of  the  proposed  drawing. 

1st.  With  one  foot  of  the  dividers  in  A,  and  any  radius,  describe  a 
circle. 

2nd.  Through  the  centre  A,  draw  a  diameter  B.  C  parallel  to  the 
sides  of  the  paper. 

3rd.  With  the  radius  from  the  points  B  and  C  lay  off  the  other 
corners  of  a  hexagon,  D.  E.  F.  G. 

4th.  Join  the  points  and  complete  the  hexagon 

5th.  From  the  centre  A>  draw  lines  to  the  alternate  corners  of  the 
hexagon,  which  will  complete  the  figure. 

The  isometrical  cube  is  a  hexahedron  supposed  to  be  viewed  at  an 
infinite  distance,  and  in  the  direction  of  the  diagonal  of  the  cube; 
in  the  diagram,  the  eye  is  supposed  to  be  placed  opposite  the 
point  A:  if  a  wire  be  run  through  the  point  A  to  the  opposite 
corner  of  the  cube,  the  eye  being  in  the  same  line,  could  only  see 
the  end  of  the  wire,  and  this  would  be  the  case  no  matter  how 
large  the  cube,  consequently  the  front  top  corner  of  the  cube  and 
the  bottom  back  corner  must  be  represented  by  a  dot,  as  at  the 
point  A.  As  the  cube  is  a  solid,  the  eye  from  that  direction  will 
see  three  of  its  sides  and  nine  of  its  twelve  edges,  and  as  the  dis- 
tance is  infinite,  all  these  edges  will  be  of  equal  length,  the  edges 
seen  are  those  shewn  in  fig.  1  by  continuous  lines;  three  of  the 
edges  and  three  of  the  sides  could  not  be  seen,  these  edges  are 
shewn  by  dotted  lines  in  fig.  1,  but  if  the  cube  were  transparent 
all  the  edges  and  sides  could  be  seen.  The  apparent  opposite 
angles  in  each  side  are  equal,  two  of  them  being  120°,  and  the 
other  two  60°;  all  the  opposite  boundary  lines  are  parallel  to  each 
other,  and  as  they  are  all  of  equal  length  may  be  measured  by 
one  common  scale,  and  all  lines  parallel  to  any  of  the  edges  of 
the  cube  may  be  measured  by  the  same  scale.  The  lines  F.  G, 
A.  C  and  E.  D  represent  the  vertical  edges  of  the  cube,  the  par- 


<>i<'  THE ISOMKTKIC.IL 


Plate  39. 


ISQMETRICAL  FIGUHKX 


.90" 


11 


Fig.  2. 


W^Ninifu 


JlbnankSons 


PLATE    XXXVIII.  77 

allelograms  Jl.  C.  D.  E  and  JL.  C.  F.  G,  represent  the  vertical 
faces  of  the  cube  and  the  parallelogram  Jl.  B.  E.  F  represents 
the  horizontal  face  of  the  cube ;  consequently,  vertical  as  well  as 
horizontal  lines  and  surfaces  may  be  delineated  by  this  method 
and  measured  by  the  same  scale,  for  this  reason  the  term  ISOME- 
TRICAL  (equally  measurable)  has  been  applied  to  this  style  of 
drawing. 


FIGURE  2 


Is  a  cube  of  the  same  size  as  fig.  1,  shaded  to  make  the  represen- 
tation more  obvious;  the  sides  of  the  small  cube  Jl,  and  the 
boundary  of  the  square  platform  on  which  the  cube  rests,  as  well 
as  of  the  joists  which  support  the  floor  of  the  platform,  are  all 
drawn  parallel  to  some  of  the  edges  of  the  cube,  and  forms  a  good 
illustration  for  the  learner  to  practice  on  a  larger  scale. 

NOTF. — A  singular  optical  illusion  may  be  witnessed  while  looking  at  this 
diagram,  if  we  keep  the  eye  fixed  on  the  point  *#,  and  imagine  the  drawing 
to  represent  the  interior  of  a  room,  the  point  A  will  appear  to  recede;  then 
if  we  again  imagine  it  to  be  a  cube  the  point  will  appear  to  advance,  and 
this  rising  and  falling  may  be  continued,  as  you  imagine  the  angle  A  to  rep- 
resent a  projecting  corner,  or  an  internal  angle. 


PLATE     XXXIX. 

EXAMPLES    IN    ISOMETRICAL    DRAWING. 


Figs.  1  and  2  are  plans  of  cubes  with  portions  cut  away.  Figs. 
3  and  4  are  isometrical  representations  of  them. 

To  draw  a  part  of  a  regular  figure,  as  in  these  diagrams,  it  is  bet- 
ter to  draw  the  whole  outline  in  pencil,  as  shewn  by  the  dotted 
lines,  and  from  the  corners  lay  off  the  indentations. 

The  circumscribing  cube  may  be  drawn  as  in  fig.  1,  Plate  38, 
with  a  radius  equal  to  the  side  of  the  plan,  or  with  a  triangle 
having  one  right  angle,  one  angle  of  60°,  and  the  cither  angle 
30°,  as  shewn  at  Jl.  Proceed  as  follows : — 

Let  B  be  the  tongue  of  a  square  or  a  straight  edge  applied  hori- 
zontally across  the  paper,  apply  the  hypothenuse  of  the  triangle 
to  the  tongue  or  straight  edge,  as  in  the  diagram,  and  draw  the 


7g  PLATE    XL. 

left  hand  inclined  lines ;  then  reverse  the  triangle  and  draw  the 
right  hand  inclined  lines ;  turn  the  short  side  of  the  triangle 
against  the  tongue  of  the  square,  and  the  vertical  lines  may  be 

drawn. 

This  instrument  so  simplifies  isometrical  drawing,  that  its  applica- 
tion is  but  little  more  difficult  than  the  drawing  of  flat  geometrical 
plans  or  elevations. 


PLATE    XL. 

EXAMPLES   IN   ISOMETRICAL  DRAWING— CONTINUED. 


Fig  1  is  the  side,  and  fig.  2  the  end  elevation  of  a  block  pierced 
through  as  shewn  in  fig  1,  and  with  the  top  chamfered  off,  as 
shewn  in  figs.  1  and  2. 

FIGURE  3. 
To  draw  the  figure  Isometrically. 

1st.  Draw  the   isometrical   lines  Jl.  B  and    C.   D;  make  Jl.  B 

equal  to  Jl.  B  fig.  1,  and  C.  D  equal  to   C.  D  fig.  2. 
2nd.  From  Jl.  B  and  D,  draw  the  vertical  lines,  and  make  them 

equal  to  B.  G,  fig.  1. 
3rd.  Draw  K.  H  and  L.  I  parallel  to  Ji.  B,  and  H.  I  and  K.  L 

parallel  to  C.  D. 

4th.  Draw  the  diagonals  H.  D  and  /.  (7,  and  through  their  inter- 
section draw  a  vertical  line  M.  G.  F.  Make  G.  F  equal  to  G. 

F,  fig.  1. 
5th.  Through  G,  draw  G.  JV,  intersecting  L.  K  in  JV*,  and  from 

JV'  draw  a  vertical  line  J\T.  E. 
6th.  Through  F,  draw  F.  E,  intersecting  JV*.  E  in  E;  then  E.  F 

represents  the  line  E.  F  in  fig.  1 . 
7th.  From  E  and  F,  lay  off  the  distances  0  and  P,  and  from   0 

and  P  draw  the  edges  of  the  chamfer  0.  K — 0.  L — P.  Hand 

P.  /,  which  complete  the  outline. 
8th.  On  Jl.  B  lay  off  the  opening  shewn  in  fig.  1,  and  from  R} 

draw  a  line  parallel  to  C.  D. 


UITIVERSIT7 


ISOMETRiaiL  FIGURES. 


K . 


M.  emeu-:. 


Fig.  1. 


a> 


PLATE    XL  79 

NOTE  1. — All  the  lines  in  this  figure,  except  the  diagonals  and  edges  of  the 
chamfer,  can  be  drawn  with  the  triangle  and  square,  as  explained  in 
Plate  39. 

NOTE  2. — All  these  lines  may  be  measured  by  the  same  scale,  except  the  in- 
clined edges  of  the  chamfer,  which  will  require  a  different  scale. 

NOTE  3. — The  intelligent  student  will  easily  perceive  from  this  figure,  how 
to  draw  a  house  with  a  hipped  roof,  placing  the  doors,  windows,  &c.,  each 
in  its  proper  place ;  or  how  to  draw  any  other  rectangular  figure.  In- 
clined lines  may  always  be  found  by  a  similar  process  to  that  we  have  pur- 
sued in  drawing  the  edges  of  the  chamfer. 

FIGURE  4 
Is  the  elevation  of  the  side  of  a  cube  with  a  large  portion  cut  out. 

FIGURE  5 


Is  the  isometrical  drawing  of  the  same,  with  the  top  of  the  cube 
also  pierced  through.  The  mode  pursued  is  so  obvious,  that  it 
requires  no  explanation :  it  is  given  as  an  illustration  for  drawing 
FURNITURE,  or  any  other  framed  object.  It  requires  but  little 
ingenuity  to  convert  fig.  5  into  the  frame  of  a  table  or  a  foot-stool. 


PLATE    XLI. 

TO    DRAW    THE    ISOMETRICAL    CIRCLE. 


FIGURE  1 

Is  the  plan  of  a  circle  inscribed  in  a  square,  with  two  diameters 
Jl.  E  and  C.  D  parallel  to  the  sides  of  the  square. 


FIGURE  2. 
To  draw  the  Isometrical  Representation. 

1st.  Draw  the  isometrical  square,  M.  JV*.  0.  P,  having  its  opposite 
angles  120°  and  60°  respectively. 
2nd.  Bisect  each  side  and  draw  JL.  B  and  C.  D. 


80  PLATE    XLI. 

3rd.  From  0  draw  0.  Jl  and  0.  D,  and  from  M  draw  M.  C  and 
M.  B  intersecting  in  Q  and  R. 

4th.  From  Q,  with  the  radius  Q.  .#,  describe  the  arc  Jl.  C}  and 
from  R,  with  the  same  radius,  describe  the  arc  D.  B. 

5th.  From  0,  with  the  radius  0.  Jl,  draw  the  arc  Jl.  D,  and  from 
M,  with  the  same  radius,  describe  C.  B,  which  completes  the  oval. 

NOTE. — An  ISOMETRICAL  PROJECTION  of  a  circle  would  be  an  ellipsis;  bat 
the  figure  produced  by  the  above  method  is  so  simple  in  its  construction  and 
approaches  so  near  to  an  ellipsis,  that  it  may  be  used  in  most  cases,  besides 
its  facility  of  construction,  its  circumference  is  so  nearly  equal  to  the  cir- 
cumference of  the  given  circle,  that  any  divisions  traced  on  the  one  may  be 
transferred  to  the  other  with  sufficient  accuracy  for  all  practical  purposes. 

FIGURE  3. 

To  divide  the   Circumference  of  the  Isometrical    Circle  into  any 
number  of  equal  parts. 

1st.  Draw  the  circle  and  a  square  around  it  as  in  fig.  2,  the  square 
may  touch  the  circle  as  in  fig.  2,  or  be  drawn  outside  as  in  fig.  3. 

2nd.  From  the  middle  of  one  of  the  sides  as  0,  erect  0.  K  per- 
pendicular to  E.  F,  and  make  0.  K  equal  to  0.  E. 

3rd.  Draw  K.  E  and  K.  F,  and  from  K  with  any  radius,  describe 
an  arc  P.  Q,  cutting  K.  E  in  P,  and  K.  F  in  Q. 

4th.  Divide  the  arc  P  4  into  one-eighth  of  the  number  of  parts  re- 
quired in  the  whole  circumference,  and  from  K,  through  these  di- 
visions, draw  lines  intersecting  E.  0  in  1,  2  and  3. 

5th.  From  the  divisions  1,  2  and  3,  in  E.  0,  draw  lines  to  the 
centre  P,  which  will  divide  the  arc  E.  0  into  four  equal  parts. 

6th.  Transfer  the  divisions  on  E.  0  from  the  corners  E.  F.  G.  H, 
and  draw  lines  to  the  centre  P,  when  the  concentric  curves  will 
be  divided  into  32  equal  parts. 

NOTE  1. — If  a  plan  of  a  circle  divided  into  any  number  of  equal  parts  be 
drawn,  as  that  of  a  cog  wheel,  the  same  measures  may  be  transferred  to  the 
isometric  curve  as  explained  in  the  note  to  fig.  2,  but  if  the  plan  be  not 
drawn,  the  divisions  can  be  made  as  in  fig.  3. 

NOTE  2. — The  term  ISOMETRICAL  PROJECTION  has  been  avoided,  as  the  pro- 
jection of  a  figure  would  require  a  smaller  scale  to  be  used  than  the  scale  to 
which  the  geometrical  plans  and  elevations  are  drawn,  but  as  the  isometri- 
cal  figure  drawn  with  the  same  scale  to  which  the  plans  are  drawn,  is  in 
every  respect  proportional  to  the  true  projection,  and  conveys  to  the  eye  the 
same  view  of  the  object,  it  is  manifestly  much  more  convenient  for  practical 
purposes  to  draw  both  to  the  same  scale. 


PLATE    XLI.  81 

NOTE  3. — In  Note  2  to  fig.  3,  Plate  40,  allusion  has  been  made  to  inclined 
lines  requiring  a  different  scale  from  any  of  the  lines  used  in  drawing  the 
isometric  cube :  for  the  mode  of  drawing  those  scales  as  well  as  for  the  further 
prosecution  of  this  branch  of  drawing,  the  student  is  referred  to  Jopling's 
and  Sopwith's  treatise  on  the  subject,  as  we  only  propose  to  give  an  intro- 
duction to  isometrical  drawing.  Sufficient,  however,  has  been  given  to  en- 
able the  student  to  apply  it  to  a  very  large  class  of  objects,  and  it  would  ex- 
tend the  size  of  this  work  too  much  (already  much  larger  than  was  intend- 
ed) if  we  pursue  the  subject  in  full. 


PERSPECTIVE, 


PLATE  XLIL 


The  design  of  the  art  of  perspective  is  to  draw  on  a  plane  surface 
the  representation  of  an  object  or  objects,  so  that  the  representa- 
tion shall  convey  to  the  eye,  the  same  image  as  the  objects  them- 
selves would  do  if  placed  in  the  same  relative  position. 

To  elucidate  this  definition  it  will  be  necessary  to  explain  the  man- 
ner in  which  the  image  of  external  objects  is  conveyed  to  the  eye. 

1st.  To  enable  a  person  to  see  any  object,  it  is  necessary  that  such 
object  should  reflect  light. 

2nd.  Light  reflected  from  a  centre  becomes  weaker  in  a  duplicate 
ratio  of  distance  from  its  source,  it  being  only  one-fourth  as  in- 
tense at  double  the  distance,  and  one-ninth  at  triple  the  distance, 
and  so  on. 

3rd.  A  ray  of  light  striking  on  any  plane  surface,  is  reflected  from 
that  surface  in  exactly  the  same  angle  with  which  it  impinges; 
thus  if  a  plane  surface  be  placed  at  an  angle  of  45°,  to  the  direc- 
tion of  rays  of  light,  the  rays  will  be  reflected  at  an  angle  of  45° 
in  the  opposite  direction.  This  fact  is  expressed  as  follows,  viz : 

THE  ANGLE  OF  REFLECTION  IS  EQUAL  TO  THE  ANGLE  OF 
_ 


82 


PLATE    XLII. 


INCIDENCE.  This  axiom,  so  short  and  pithy,  should  be  stored 
in  the  memory  with  some  others  that  we  propose  to  give,  to  be 
brought  forward  and  applied  whenever  required. 

4th.  Rays  of  light  reflected  from  a  body  proceed  in  straight  lines 
until  interrupted  by  meeting  with  other  bodies,  which  by  reflec- 
tion or  refraction,  change  their  direction. 

5th.  REFRACTION  of  LIGHT.  When  a  ray  of  light  passes  from  a 
rare  to  a  more  dense  medium,  as  from  a  clear  atmosphere  through 
a  fog  or  from  the  air  into  water,  it  is  bent  out  of  its  direct  course: 
thus  if  we  thrust  a  rod  into  water,  it  appears  broken  or  bent  at  the 
surface  of  the  water ;  objects  have  been  seen  through  a  fog  by  the 
bending  of  the  rays,  that  could  not  possibly  be  seen  in  clear  wea- 
ther; this  bending  of  the  rays  of  light  is  called  refraction,  and  the 
rays  are  said  to  be  refracted  :  this-  effect,  (produced  however  by  a 
different  cause)  may  often  be  seen  by  looking  through  common 
window^  glass,  when  in  consequence  of  the  irregularities  of  its 
surface,  the  view  of  objects  without  is  much  distorted. 

6th.  A  portion  of  light  is  absorbed  by  all  bodies  receiving  it  on 
their  surface,  consequently  the  amount  of  light  reflected  from  an 
object  is  not  equal  to  the  quantity  received. 

7th.  The  amount  of  absorption  is  not  the  same  in  all  bodies,  but 
depends  on  the  color  and  quality  of  the  reflecting  surface ;  if  a  ray 
falls  on  the  bright  polished  surface  of  a  looking-glass,  most  of  it 
will  be  reflected,  but  if  it  should  fall  on  a  surface  of  black  cloth, 
most  of  it  would  be  absorbed.  White  or  light  colors  reflect  more 
of  a  given  ray  of  light  than  dark  colors ;  polished  surfaces  reflect 
more  than  those  which  are  unpolished,  and  smooth  surfaces  more 
than  rough. 

8th.  As  all  objects  absorb  more  or  less  light,  it  follows  that  at  each 
reflection  the  ray  will  become  weaker  until  it  is  no  longer  per- 
ceptible. 

9th.  Rays  received  from  a  luminous  source  are  called  direct,  and 
the  parts  of  an  object  receiving  these  direct  rays  are  said  to  be  in 
LIGHT.  The  portions  of  the  surface  so  situated  as  not  to  receive 
the  direct  rays  are  said  to  be  in  SHADE  ;  if  the  object  receiving 
the  direct  rays  is  opaque,  it  will  prevent  the  rays  from  passing  in 
that  direction,  and  the  outline  of  its  illuminated  parts  will  be  pro- 
jected on  the  nearest  adjoining  surface :  the  figure  so  projected  is 
called  its  SHADOW. 
10th.  The  parts  of  an  object  in  shade  will  always  be  lighter  than 
the  shadow,  as  the  object  receives  more  or  less  reflected  light  from 


PLATE    XLII.  83 

the  atmosphere  and  adjoining  objects,  the  quantity  depending  on 
the  position  of  the  shaded  surface,  and  on  the  position  and  quali- 
ty of  the  surrounding  objects. 

1 1th.  If  an  object  were  so  situated  as  to  receive  only  a  direct  ray 
of  light,  without  receiving  reflected  light  from  other  sources,  the 
illuminated  portion  could  alone  be  seen;  but  for  this  universal  law 
of  reflection  we  should  be  able  to  see  nothing  that  is  not  illumi- 
nated by  the  direct  rays  of  the  sun  or  by  some  artificial  means,  and 
all  beyond  would  be  one  gloomy  blank. 

12th.  Rays  of  light  proceeding  in  straight  lines  from  the  surfaces 
of  objects,  meet  in  the  front  of  the  eye  of  the  spectator  where 
they  cross  each  other,  and  form  an  inverted  image  on  the  back  of 
the  eye,  of  all  objects  within  the  scope  of  vision. 

13th.  The  size  of  the  image  so  formed  on  the  retina  depends  on 
the  size  and  distance  of  the  original ;  the  shape  of  the  image  de- 
pends on  the  angle  at  which  it  is  seen. 

NOTE. — The  size  of  objects  diminishes  directly  as  the  distance  increases,  ap- 
pearing at  ten  times  the  distance,  only  a  tenth  part  as  large ;  the  knowledge 
of,this  fact  has  produced  a  system  of  arithmetical  perspective,  which  enables 
the  draughtsman  to  proportion  the  sizes  of  objects  by  calculation. 

14th.  The  strength  of  the  image  depends  on  the  degree  of  illumi- 
nation of  the  original,  and  on  its  distance  from  the  eye,  objects 
becoming  more  dim  as  they  recede  from  the  spectator. 

15th.  To  give  a  better  idea  of  the  operation  of  the  eye  in  viewing 
an  object,  let  us  refer  to  fig.  1.  The  circle  Jl  is  intended  to  repre- 
sent a  section  of  the  human  eye,  H  the  pupil  in  front,  K  the 
crystalline  lens  in  which  the  rays  are  all  converged  and  cross  each 
other,  and  M  the  concave  surface  of  the  back  of  the  eye  called 
the  retina,  on  which  the  image  is  projected. 

16th.  Let  us  suppose  the  eye  to  be  viewing  the  cross  B.  C,  and 
that  the  parallelogram  N.  0.  P.  Q  represents  a  picture  frame 
in  which  a  pane  of  glass  is  inserted;  the  surface  of  the  glass  slight- 
ly obscured  so  as  to  allow  objects  to  be  traced  on  it,  then  rays 
from  every  part  of  the  cross  will  proceed  in  straight  lines  to  the 
eye,  and  form  the  inverted  image  C.  B  on  the  retina.  If  with  a 
pencil  we  were  to  trace  the  form  of  the  cross  on  the  glass  so  as  to 
interrupt  the  view  of  the  original  object,  we  should  have  a  true 
perspective  representation  of  the  original,  which  would  form  ex- 
actly the  same  sized  image  on  the  retina;  thus  the  point  b  would 
intercept  the  view  of  B,  c  of  (7,  d  of  D  and  e  of  E,  and  if  colored 
the  same  as  the  original,  the  image  formed  from  it  would  be  the 
same  in  every  respect  as  from  the  original. 


84  PLATE    XLII. 

17th.  If  we  move  the  cross  B.  C  to  F.  G,  the  image  formed  on 
the  retina  would  be  much  larger,  as  shewn  at  G.  Fy  and  the  rep- 
resentation on  the  glass  would  be  larger,  the  ray  from  F  passing 
through/,  and  the  ray  from  G  passing  through  g,  shewing  that 
the  same  object  will  produce  a  larger  or  smaller  image  on  the  re- 
tina as  it  advances  to  or  recedes  from  the  spectator;  the  farther  it 
recedes,  the  smaller  will  be  the  image  formed,  until  it  becomes  so 
small  as  to  be  invisible. 

18th.  Fig.  2  is  given  to  elucidate  the  same  subject.  If  we  suppose  a 
person  to  be  seated  in  a  room,  the  ground  outside  to  be  on  a  level 
with  the  bottom  of  the  window  JL.  B,  the  eye  at  S  in  the  same 
level  line,  and  a  series  of  rods  C.  D.  E.  F  of  the  same  height 
of  the  window  to  be  planted  outside,  the  window  to  be  filled' 
with  four  lights  of  glass  of  equal  size,  then  the  ray  from  the  bot- 
tom of  all  the  rods  would  pass  through  the  bottom  of  the  window; 
the  ray  from  the  top  of  C  would  pass  through  the  top  of  the  win- 
dow; from  the  top  of  D  a  little  farther  off,  it  would  pass  through 
the  third  light;  the  ray  from  E  would  pass  through  the  middle, 
and  F  would  only  occupy  the  height  of  one  pane. 

19th.  Fig.  3.  Different  sized  and  shaped  objects  may  produce  the 
sanie  image;  thus  the  bent  rods  JL  and  C,  and  the  straight  rods 
B  and  D  would  produce  the  same  image,  being  placed  at  different 
distances  from  the  eye,  and  all  contained  in  the  same  angle  D.  S. 
E.  As  the  bent  rods  JL  and  C  are  viewed  edgewise  they  would 
form  the  same  shaped  image  as  if  they  were  straight.  The  angle 
formed  by  the  rays  of  light  passing  from  the  top  and  bottom  of 
an  object  to  the  eye,  as  D.  S.  E,  is  called  the  VISUAL  ANGLE,  and 
the  object  is  said  to  subtend  an  angle  of  so  many  degrees,  measur- 
ing the  angle  formed  at  S. 

20th.  Of  FORESHORTENING.  When  an  object  is  viewed  obliquely 
it  appears  much  shorter  than  if  its  side  is  directly  in  front  of  the 
eye ;  if  for  instance  we  hold  a  pencil  sidewise  at  arms  length  op- 
posite the  eye,  we  should  see  its  entire  length;  then  if  we  incline 
the  pencil  a  little,  the  side  will  appear  shorter,  and  one  of  the 
ends  can  also  be  seen,  and  the  more  the  pencil  is  inclined  the 
smaller  will  be  the  angle  subtended  by  its  side,  until  nothing  but  the 
end  would  be  visible.  Again  if  a  wheel  be  placed  perpendicular- 
ly opposite  the  eye,  its  rim  and  hub  would  shew  perfect  circles, 
arid  the  spokes  would  all  appear  to  be  of  the  same  length,  but  if  we 
incline  the  wheel  a  little,  the  circles  will  appear  to  be  ellipses, 
and  the  spokes  appear  of  different  lengths,  dependant  on  the  an- 


PLATE    XLI1.  85 

gle  at  which  they  are  viewed;  the  more  the  wheel  is  inclined  the 
shorter  will  be  the  conjugate  diameter  of  the  ellipsis,  until  the 
whole  would  form  a  straight  line  whose  length  would  be  equal 
to  the  diameter,  and  its  breadth  equal  to  the  thickness  of  the 
wheel.  This  decrease  of  the  angle  subtended  by  an  object,  when 
viewed  obliquely,  is  called  foreshortening. 


PLATE   XLIII. 


FIGURE  1. 

21st.  If  we  suppose  a  person  to  be  standing  on  level  ground,  with 
his  eye  at  S,  the  line  JL.  F  parallel  to  the  surface  and  about  five 
feet  above  it,  and  the  surface  G.  E  to  be  divided  off  into  spaces 
of  five  feet,  as  at  B.  C.  D  and  E,  then  if  from  S,  with  a  radius  S. 
G,  we  describe  the  arc  Jl.  G,  and  from  the  points  B.  C.  D  and 
E  we  draw  lines  to  £,  cutting  the  arc  in  H.  K.  L  and  M,  the 
distances  between  the  lines  on  the  arc,  will  represent  the  angle 
subtended  in  the  eye  by  each  space,  and  if  we  adopt  the  usual 
mode  for  measuring  an  angle,  and  divide  the  quadrant  into  90°, 
it  will  be  perceived  that  the  first  space  of  five  feet  subtends  an 
angle  of  45°,  equal  to  one-half  of  the  angle  that  would  be  sub- 
tended by  a  plane  that  would  extend  to  the  extreme  limits  of  vi- 
sion; the  next  space  from  B  to  C  subtends  an  angle  of  about 
18  1-2°,  from  C  to  D  about  8°,  and  from  D  to  E  about  4  1-2°,  and 
the  angle  subtended  would  constantly  become  less,  until  the  divi- 
sions of  the  spaces  would  at  a  short  distance  appear  to  touch  each 
other,  a  space  of  five  feet  subtending  an  angle  so  small,  that  the  eye 
could  not  appreciate  it.  It  is  this  foreshortening  that  enables  us 
in  some  measure  to  judge  of  distance. 

22nd.  If  instead  of  a  level  plane,  the  person  at  *S>  be  standing  at  the 
foot  of  a  hill,  the  surface  being  less  inclined  would  diminish  less 
rapidly,  but  if  on  the  contrary  he  be  standing  on  the  brow  of  a 
hill  looking  downward,  it  would  diminish  more  rapidly;  hence  we 
derive  the  following  axiom:  THE  DEGREE  OF  FORESHORTENING 

OF    OBJECTS     DEPENDS    ON    THE    ANGLE    AT     WHICH     THEY    ARE 
VIEWED 


86  PLATE    XLII1. 

23rd.  PERSPECTIVE  may  be  divided  into  two  branches,  LINEAR 
and  AERIAL. 

24th.  LINEAR  PERSPECTIVE  teaches  the  mode  of  drawing  the  lines 
of  a  picture  so  as  to  convey  to  the  eye  the  apparent  SHAPE  or 
FIGURE  of  each  object  from  the  point  at  which  it  is  viewed. 

25th.  AERIAL  PERSPECTIVE  teaches  the  mode  of  arranging  the 
direct  and  reflected  LIGHTS,  SHADES,  and  SHADOWS  of  a  picture, 
so  as  to  give  to  each  part  its  requisite  degree  of  tone  and  color, 
diminishing  the  strength  of  each  tint  as  the  objects  recede,  until 
in  the  extreme  distance,  the  whole  assumes  a  bluish  gray  which  is 
the  color  of  the  atmosphere.  This  branch  of  the  art  is  requisite 
to  the  artist  who  would  paint  a  landscape,  and  can  be  better  learnt 
by  the  study  of  nature  and  the  paintings  of  good  masters,  than  by 
any  series  of  rules  which  would  require  to  be  constantly  varied. 

26th.  Linear  perspective,  on  the  contrary,  is  capable  of  strict  mathe- 
matical demonstration,  and  its  rules  must  be  positively  followed  to 
produce  the  true  figure  of  an  object. 


DEFINITIONS. 

27th.  The  PERSPECTIVE  PLANE  is  the  surface  on  which  the  pic- 
ture is  drawn,  and  is  supposed  to  be  placed  in  a  vertical  position 
between  the  spectator  and  the  object — thus  in  fig.  1,  Plate  42,  the 
parallelogram  i/V*.  0.  P.  Q  is  the  perspective  plane. 

28th.  The  GROUND  LINE  or  BASE  LINE  of  a  picture  is  the  seat  of 
the  perspective  plane,  as  the  line  Q.  P,  fig.  1,  Plate  42,  and 
G.  L,  fig.  2,  Plate  43. 

29th.  The  HORIZON.  The  natural  horizon  is  the  line  in  which  the 
earth  and  sky,  or  sea  and  sky  appear  to  meet;  the  horizon  in  a 
perspective  drawing  is  at  the  height  of  the  eye  of  the  spectator.  If 
the  object  viewed  be  on  level  ground,  the  horizon  will  be  about 
five  feet  or  five  and  a  half  feet  above  the  ground  line,  as  it  is  repre- 
sented by  V.  L}  fig.  2.  If  the  spectator  be  viewing  the  object 
from  an  eminence, the  horizon  will  be  higher,  and  if  the  spectator 
be  lower  than  the  ground  on  which  the  object  stands,  the  horizon 
will  be  lower;  thus  the  horizon  in  perspective,  means  the  height 
of  the  eye  of  the  spectator,  arid  as  an  object  may  be  viewed  by  a 
person  reclining  on  the  ground,  or  standing  upright  on  the 
ground,  or  he  may  be  elevated  on  a  chair  or  table,  it  follows  that 
the  horizon  may  be  made  higher  or  lower,  at  the  pleasure  of  the 


PLATE    XLIII.  87 

draughtsman;  but  in  a  mechanical  or  architectural  view  of  a  de- 
sign, it  should  be  placed  about  five  feet  above  the  ground  line. 

NOTE. — The  tops  of  all  horizontal  objects  that  are  below  the  horizon,  and 
the  under  sides  of  objects  above  the  horizon,  will  appear  more  or  less  dis- 
played as  they  recede  from  or  approach  to  the  horizon. 

30th.  The  STATION  POINT,  or  POINT  of  VIEW  is  the  position  of  the 
spectator  when  viewing  the  object  or  picture. 

31st.  The  POINT  of  SIGHT.  If  the  spectator  standing  at  the  sta- 
tion point  should  hold  his  pencil  horizontally  at  the  level  of  his 
eye  in  such  a  position  that  the  end  only  could  be  seen,  it  would 
cover  a  small  part  of  the  object  situated  in  the  horizon ;  this  point 
is  marked  as  at  S,  fig.  2,  and  called  the  point  of  sight.  It  must  be 
remembered  that  the  point  of  sight  is  not  the  position  of  a  specta- 
tor when  viewing  an  object;  but  a  point  in  the  horizon  directly 
opposite  the  eye  of  the  spectator,  and  from  which  point  the  spec- 
tator may  be  at  a  greater  or  less  distance. 

32nd.  POINTS  of  DISTANCE  are  set  off  on  the  horizon  on  either 
side  of  the  point  of  sight  as  at  D.  D',  and  represent  the  distance 
of  the  spectator  from  the  perspective  plane.  As  an  object  may  be 
viewed  at  different  distances  from  the  perspective  piane,  it  fol- 
lows that  these  points  may  be  placed  at  any  distance  from  the 
point  of  sight  to  suit  the  judgment  of  the  draughtsman,  but  they 
should  never  be  less  than  the  base  of  the  picture. 

NOTE  1. — Although  the  height  of  the  horizon,  and  the  points  of  distance  may 
be  varied  at  pleasure,  it  is  only  from  that  distance  and  with  the  eye  on  a  le- 
vel with  the  horizon  that  a  picture  can  be  viewed  correctly. 

NOTE  2. — In  the  following  diagrams  the  points  of  distance  have  generally 
been  placed  within  the  boundary  of  the  plates,  as  it  is  important  that  the 
learner  should  see  the  points  to  which  the  lines  tend ;  they  should  be  copied 
with  the  points  of  distance  much  farther  off. 

33rd.  VISUAL  RAYS.  All  lines  drawn  from  the  object  to  the  eye 
of  the  spectator  are  called  visual  rays. 

34th.  The  MIDDLE  RAY,  or  CENTRAL  VISUAL  RAY  is  a  line  pro- 
ceeding from  the  eye  of  the  spectator  to  the  point  of  sight;  exter- 
nal visual  rays  are  the  rays  proceeding  from  the  opposite  sides 
of  an  object,  or  from  the  top  and  bottom  of  an  object  to  the  eye. 
The  angle  formed  in  the  eye  by  the  external  rays,  is  called  the 
visual  angle. 

NOTE. — The  perspective  plane  must  always  be  perpendicular  to  the  middle 
visual  ray. 

35th.  VANISHING  POINTS.  It  has  been  shewn  at  fig.  1  in  this 
plate  that  objects  of  the  same  size  subtend  a  constantly  decreasing 


88  PLATE    XLIII. 

angle  in  the  eye  as  they  recede  from  the  spectator,  until  they  are 
no  longer  visible ;  the  point  where  level  objects  become  invisible 
or  appear  to  vanish,  will  always  be  in  the  horizon,  and  is  called 
the  vanishing  point  of  that  object. 

36th.  The  POINT  of  SIGHT  is  called  the  PRINCIPAL  VANISHING 
POINT,  because  all  horizontal  objects  that  are  parallel  to  the  mid- 
dle visual  ray  will  vanish  in  that  point.  If  we  stand  in  the  mid- 
dle of  a  street  looking  directly  toward  its  opposite  end  as  in  Plate 
54,  (the  Frontispiece,)  all  horizontal  lines,  such  as  the  tops  and 
bottoms  of  the  doors  and  windows,  eaves  and  cornices  of  the 
houses,  tops  of  chimnies,  &c.  will  tend  toward  that  point  to  which 
the  eye  is  directed,  and  if  the  lines  were  continued  they  would 
unite  in  that  point.  Again,  if  we  stand  in  the  middle  of  a  room 
looking  towards  its  opposite  end,  the  joints  of  the  floor,  corners 
of  ceiling,  washboards  and  the  sides  of  furniture  ranged  against 
the  side  walls,  or  placed  parallel  to  them,  would  all  tend  to  a  point 
in  the  end  of  the  room  at  the  height  of  the  eye. 

37th.  The  VANISHING  POINTS  of  horizontal  objects  not  parallel 
with  the  middle  ray  will  be  in  some  point  of  the  horizon,  but  not 
in  the  point  of  sight.  These  vanishing  points  are  called  acciden- 
tal points. 

38th.  DIAGONALS.  Lines  drawn  from  the  perspective  plane  to 
the  point  of  distance  as  JY.  Df  and  0.  Z>,  or  from  a  ray  drawn  to 
the  point  of  sight  as  E.  D'  and  F.  D,  are  called  diagonals;  all  such 
lines  represent  lines  drawn  at  an  angle  of  45°  to  the  perspective 
plane,  and  form  as  in  this  figure  the  diagonals  of  a  square,  whose 
side  is  parallel  to  the  perspective  plane. 

39th.  Of  VANISHING  PLANES.  On  taking  a  position  in  the  mid- 
dle of  a  street  as  described  in  paragraph  36,  it  is  there  stated  that 
all  lines  will  tend  to  a  point  in  the  distance  at  the  height  of  the 
eye,  called  the  point  of  sight,  or  principal  vanishing  point;  this  is 
equally  true  of  horizontal  or  vertical  planes  that  are  parallel  to  the 
middle  visual  ray :  for  if  we  suppose  the  street  between  the  curb 
stones,  and  the  side  walks  of  the  street  to  be  three  parallel  hori- 
zontal planes  as  m  Plate  54,  their  boundaries  will  all  tend  to 
the  vanishing  point,  until  at  a  distance,  depending  on  the  breadth 
of  the  plane,  they  become  invisible.  Again,  the  walls  of  the  houses 
on  both  sides  of  the  street  are  vertical  planes,  bounded  by  the 
eaves  of  the  roofs  and  by  their  intersection  with  the  horizontal 
planes  of  the  side  walks,  these  boundaries  would  also  tend  to  the 
same  point,  and  if  the  rows  of  houses  were  continued  to  a  suffi- 


Plate  42. 


PERSPECTIVE. 


Fig.  2. 


Fig.  3. 


WT^Minifte 


Plate  43. 


PERSPECTIVE . 


Fig.l. 


/'/>/.  2. 


1,5 


PLATE    XLIII.  89 

cient  distance,  these  planes  would  vanish  in  the  same  point;  if 
the  back  walls  of  the  houses  are  parallel  to  the  front,  the  planes 
formed  by  them  will  vanish  in  the  same  point,  and  if  any  other 
streets  should  be  running  parallel  to  the  first,  their  horizontal  and 
vertical  planes  would  all  tend  to  the  same  point. 

J^OTE. — A  BIRD'S  EYE  VIEW  of  the  streets  of  a  town  laid  out  regularly,  would 
fully  elucidate  the  truth  of  the  remarks  in  this  paragraph.  When  the  horizon 
of  a  picture  is  placed  very  high  above  the  tops  of  the  houses,  as  if  the  spec- 
tator were  placed  on  some  very  elevated  object,  or  if  seen  as  a  bird  would 
see  it  when  on  the  wing,  the  view  is  called  a  bird's  eye  view;  in  a  represen- 
tation of  this  kind  the  tops  of  all  objects  are  visible,  and  the  tendency  of  all 
the  planes  and  lines  parallel  to  the  middle  visual  ray  to  vanish  in  the  point 
of  sight,  is  very  obvious. 

40th.  If  we  were  viewing  a  room  as  described  in  paragraph  36, 
the  ceiling  and  floor  would  be  horizontal  planes,  and  the  walls 
vertical  planes,  and  if  extended  would  all  vanish  in  the  point  of 
sight;  or  if  we  were  viewing  the  section  of  a  house  of  several 
stories  in  height,  all  the  floors  and  ceilings  would  be  horizontal 
planes,  and  all  the  parallel  partitions  and  walls  would  be  vertical 
planes,  and  would  all  vanish  in  the  same  point. 

41st.  When  the  BOUNDARIES  OF  INCLINED  PLANES  are  horizontal 
lines  parallel  to  the  middle  ray,  the  planes  will  vanish  in  the  point 
of  sight;  thus  the  roofs  of  the  houses  in  Plate  54,  bounded  by 
the  horizontal  lines  of  the  eaves  and  ridge,  are  inclined  planes 
vanishing  in  the  point  of  sight. 

42nd.  PLANES  PARALLEL  TO  THE  PLANE  OF  THE  PICTURE  have 
no  vanishing  point,  neither  have  any  lines  drawn  on  such  planes. 

43rd.  VERTICAL  OR  HORIZONTAL  PARALLEL  PLANES  running  at 
any  inclination  to  the  middle  ray  or  perspective  plane,  vanish  in 
accidental  points  in  the  horizon,  as  stated  in  paragraph  37 ;  as  for 
example,  the  walls  and  bed  of  a  street  running  diagonally  to  the 
plane  of  the  picture,  or  of  a  single  house  as  in  Plate  53,  where  the 
opposite  sides  vanish  in  accidental  points  at  different  distances  from 
the  point  of  sight,  because  the  walls  form  different  angles  with  the 
perspective  plane,  as  shewn  by  the  plan  of  the  walls  B.  D  and  D. 
C,  fig.  1. 

44th.  ALL  HORIZONTAL  LINES  DRAWN  ON  A  PLANE,  or  running 
parallel  to  a  plane,  vanish  in  the  same  point  as  the  plane  itself. 

45th.  INCLINED  LINES  vanish  in  points  perpendicularly  above  or 
below  the  vanishing  point  of  the  plane,  and  if  they  form  the  same 
angle  with  the  horizon  in  different  directions  as  the  gables  of  the 


12 


90  PLATE    XLIII. 

house  in  fig.  2,  Plate  53,  the  vanishing  points  will  be  equidistant 

from  the  horizon. 
From  what  has  been  said  we  derive  the  following  AXIOMS;  their 

importance  should  induce  the  student  to  fix  them  well   in  his 

memory : 
1st.  The  ANGLE  OF  REFLECTION  OF  LIGHT  is  equal  to  the  angle 

of  incidence.     See  paragraph  No.  3,  page  81. 
2nd.  The  SHADOW  of  an  object  is  always  darker  than  the  object 

itself.     See  paragraph  10,  page  82. 
3rd.  The  DEGREE  OF  FORESHORTENING  of  objects  depends  on  the 

angle  at  which  they  are  viewed.     See  paragraph  20,  page  84. 
4th.  The  APPARENT  SIZE  of  an  object  decreases  exactly  as  its  dis- 
tance from  the  spectator  is  increased.  See  paragraph  35,  p.  87. 
5th.  PARALLEL  PLANES  and  LINES  vanish  to    a  common  point. 

See  paragraph  36,  page  88. 
6th.  ALL  PARALLEL  PLANES  whose  boundaries  are  parallel  to  the 

middle  visual  ray,  vanish  in  the  point  of  sight.     See  paragraph 

36,  page  88. 
7th.  ALL  HORIZONTAL  LINES  parallel  to  the  middle  ray  vanish  in 

the  point  of  sight. 
8th.  HORIZONTAL  LINES  AT  AN  ANGLE  OF  45°  with  the  plane  of 

the  picture,  vanish  in  the  points  of  distance.     See  paragraph  38, 

page  88. 

9th.  PLANES  AND  LINES  PARALLEL  TO  THE  PLANE  OF  THE  PIC- 
TURE have  no  vanishing  point. 


PRACTICAL    PROBLEMS. 

1st.  To  draw  the  perspective  representation  of  the  square  N.  O.  P. 
Q,  viewed  in  the  direction  of  the  line  W.  B,  with  one  of  its  sides 
N.  O  touching  the  perspective  plane  G.  L,  and  parallel  with  it. 

1st.  Draw  the  horizontal  line  V.  L  at  the  height  of  the  eye, 

2nd.  From  C,  the  centre  of  the  side  JV*.  0,  draw  a  perpendicular 
to  V.  L,  cutting  it  in  S.  Then  S  is  the  point  of  sight  or  the  prin- 
cipal vanishing  point,  and  C.  S  the  middle  visual  ray. 

3rd.  As  the  sides  JV.  P  and  0.  Q  are  parallel  to  the  middle  ray 
C.  S,  they  will  vanish  in  the  point  of  sight.  Therefore  from  JV 
and  0  draw  rays  to  S;  these  are  the  external  visual  rays. 

4th.  From  S,  set  off  the  points  of  distance  D.  Dr  at.  pleasure,  equi- 
distant from  S,  and  from  JV  and  0,  draw  the  diagonals  J\T.  D'  and 


PLATE    XLIII.  91 

0.  D.     Then  the  intersection  of  these  diagonals  with  the  external 
visual  rays  determine  the  depth  of  the  square. 
5th.  Draw  E.  F  parallel  to  JV*.  0.  Then  the  trapezoid  JV.  0.  E.  F 
is  the  perspective  representation  of  the  given  square  viewed  at  a 
distance  from  W  on  the  line  W.  B,  equal  to  S.  D. 

2nd.    To   draw  the  Representation  of  another  Square  of  the  same 
size  immediately  in  the  rear  of  E.  F. 

1st.  From  E,  draw  E.  H,  intersecting  0.  S  in  H,  and  from  F, 
draw  F.  D,  intersecting  JV.  S  in  B. 

2nd.  Draw  B.  H  parallel  to  E.  F,  which  completes  the  second 
square ;  and  the  trapezoid  JV.  0.  H.  B  is  the  representation  of  a 
parallelogram  whose  side  0.  H  is  double  the  side  of  the  given 
square. 

NOTE*. — If  from  W  on  the  line  W.  B  we  set  off  the  distance  S.  D,  extending 
in  the  example  outside  of  the  plate,  (which  represents  the  distance  from  which 
the  picture  is  viewed,)  and  from  JVand  0  draw  rays  to  the  point  so  set  off, 
cutting  P.  Q  in  R  and  T,  then  the  lines  R.  T  and  E.  F  will  be  of  equal 
length,  and  prove  the  correctness  of -the  diagram. 


PLATE    XLIV. 


FIGURE  1. 

To  draw  a  Perspective  Plan  of  a  Square  and  divide  it  into  a  given 
number  of  Squares,  say  sixty-four. 

Let  G.  L  be  the  base  line,  V.  L  the  horizon,  S  the  point  of  sight, 
and  JV.  0  the  given  side  of  the  square. 

1st.  From  JVand  0,draw  rays  to  S  and  diagonals  to  D.  D,  inter- 
secting each  other  in  P  and  Q,  draw  P.  Q. 

2nd.  Divide  JV".  0  into  eight  equal  parts,  and  from  the  points  of 
division  draw  rays  to  S. 

3rd.  Through  the  points  of  intersection  formed  with  those  rays  by 
the  diagonals,  draw  lines  parallel  to  JV.  0,  which  will  divide  the 
square  as  required,  and  may  represent  a  checker  board  or  a 
pavement  of  square  tiles. 


92  PLATE  XLIV. 

OF  HALF  DISTANCE. 

When  the  points  of  distance  are  too  far  off  to  be  used  convenient- 
ly, half  the  distance  may  be  used;  as  for  example,  if  we  bisect  S. 
D  in  1-2  D,  and  JV.  0  in  C,  and  draw  a  line  from  C  to  1-2  D, 
it  will  intersect  JV.  £  in  P,  being  in  the  same  point  as  by  the  diago- 
nal drawn  from  0  the  opposite  side  of  the  square,  to  the  whole 
distance  at  D. 

NOTE. — Any  other  fraction  of  the  distance  may  be  used,  provided  that  the 
divisions  on  the  base  line  be  measured  proportionately. 

tfWWWWWVVWWMWMVtt 

FIGURE  2. 


To  draw  the  Plan  of  a  Room  with  Pilasters  at  its  sides,  the  base 
line,  horizon,  point  of  sight,  and  points  of  distance  given. 

NOTE. — To  AVOID  REPETITIONS,  in  the  following  diagrams  we  shall  suppose 
the  base  line,  the  horizon  V.  L,  the  point  of  sight  S,  and  the  points  of 
distance  D.  D  to  be  given. 

1st.  Let  J]T.  0  be  the  width  of  the  proposed  room,  then  draw  JV". 
S  and  0.  S  representing  the  sides  of  the  room. 

2nd.  From  JV*  toward  0  lay  down  the  width  of  each  pilaster,  and 
the  spaces  between  them,  and  draw  lines  to  D,  then  through  the 
points  where  these  lines  intersect  the  external  visual  ray  JV".  S, 
draw  lines  parallel  with  N.  0  to  the  line  0.  S. 

3rd.  From  JV*  and  0,  set  off  the  projection  of  the  pilasters  and 
draw  rays  to  the  point  of  sight.  The  shaded  parts  shew  the  posi- 
tion of  the  pilasters. 

4th.  If  from  N  we  lay  off  the  distances  and  widths  of  the  pilasters 
toward  M,  and  draw  diagonals  to  the  opposite  point  of  distance, 
N.  S  would  be  intersected  in  exactly  the  same  points. 

NOTE. — Any  rectangular  object  may  be  put  in  perspective  by  this  method, 
without  the  necessity  of  drawing  a  geometrical  plan,  as  the  dimensions  may 
all  be  laid  off  on  the  ground  line  by  any  scale  of  equal  parts. 


FIGURE  3. 


To  shorten  the  depth  of  a  perspective  drawing,  thereby  producing 
the  same  effect  as  if  the  points  of  distance  were  removed  much 
farther  off. 

1st.  Let  all  the  principal  lines  be  given  as  above,  and  the  pilasters 
and  spaces  laid  off  on  the  base  line  from  N. 


U1U7BRSITT 


Fig.l. 


Fig.  3. 


r*  


Tesselated  Pavements 
Fit/.l. 


Fig.  3. 


PLATE    XLIV.  93 

2nd.  From  the  dimensions  on  the  base  line  draw  diagonals  to  the 
point  of  distance  D.  The  diagonal  from  M  the  outside  pilaster 
will  intersect  JV.  S  in  P. 

3rd.  From  N  erect  a  perpendicular  JV*.  B  to  intersect  the  diago- 
nals, and  from  those  intersections  draw  horizontal  lines  to  inter- 
sect N.  S. 

4th.  If  from  N  we  draw  the  inclined  line  N.  E  and  transfer  the 
intersections  from  it  to  V.  0,  it  will  reduce  the  depth  much  more, 
as  shown  at  0.  S.  • 

Most  of  the  foregoing  diagrams  may  be  drawn  as  well  with  one 
point  of  distance  as  with  two. 


PLATE     XLV. 

TESSELATED    PAVEMENTS. 


FIGURE  1. 


To  draw  a  pavement  of  square  tiles,  with  their  sides  placed  diago- 
nally to  the  perspective  plane. 

1st.  Draw  the  perspective  square  JV*.  0.  P.  Q. 

2nd.  Divide  the  base  line  JV.  0  into  spaces  equal  to  the  diagonal 

of  the  tiles. 
3rd.   From  the  divisions  on  JV.  0  draw  diagonals  to  the  points  of 

distance. 
4th.  Tint  every  alternate  square  to  complete  the  diagram. 


FIGURE  2. 

To  draw  a  pavement  of  square  black  tiles  with  a  white  border  around 
them,  the  sides  of  the  squares  parallel  to  the  perspective  plane  and 
middle  visual  ray. 

1st.  Draw  the  perspective  square,  and  divide  X.  0  into  alternate 
spaces  equal  to  the  breadth  of  the  square  and  borders. 

2nd.  From  the  divisions  on  X.  0  draw  rays  to  the  point  of  sight, 
and  from  X  draw  a  diagonal  to  the  point  of  distance. 


94 


PLATE    XLV. 


3rd.  Through  the  intersections  formed  by  the  diagonal,  with  the 
rays  drawn  from  the  divisions  on  X.  0,  draw  lines  parallel  to  A". 
0,  to  complete  the  small  squares. 


FIGURE  3. 


To  draw  a  Pavement  composed  of  Hexagonal  and  Square  Blocks. 

1st.  Divide  the  diameter  of  one  of  the  proposed  hexagons  a.  b  into 
three  equal  parts,  and  from  the  points  of  division  draw  rays  to 
the  point  of  sight. 

2nd.  From  a,  draw  a  diagonal  to  the  point  of  distance,  and  through 
the  intersections  draw  the  parallel  lines. 

3rd.  From  1,  2,  3  and  4,  draw  diagonals  to  the  opposite  points  of 
distance,  which  complete  the  hexagon. 

4th.  Lay  off  the  base  line  from  a  and  b  into  spaces  equal  to  one- 
third  of  the  given  hexagon,  and  draw  rays  from  them  to  the  point 
of  sight;  then  draw  diagonals  as  in  the  diagram,  to  complete  the 
pavement. 


PLATE    XLVI. 


FIGURE   1. 


To  draw  the  Double  Square  E.  F.  G.  H,  viewed  diagonally,  with 
one  of  its  corners  touching  the  Perspective  Plane. 

1st.  Prolong  the  sides  of  the  squares  as  shewn  by  the  dotted  lines 

to  intersect  the  perspective  plane. 
2nd.  From  the  points  of  intersection,  draw  diagonals  to  the  points 

of  distance,  their  intersections  form  the  diagonal  squares. 
3rd.  The  square  Jl.  B.  N.  0  is  drawn  around  it  on  the  plan  and 

also  in  perspective,  to  prove  that  the  same  depth  and  breadth  is 

given  to  objects  by  both  methods  of  projection. 


THE 

UF17BRSITr 


/'/<//<> 


/'If/./. 


fi'f/.  2. 

s 


Plate  4  7. 


LINE  or    ELEVATION. 


/vy./. 


Fig.  2. 


5  4  3  2  1 


1  PLATE  XLVI.  95 

FIGURE  2. 

To  draw  the  Perspective  Representation  of  a  Circle  mewed  directly 
in  front  and  touching  the  Perspective  Plane. 

Find  the  position  of  any  number  of  points  in  the  Curve. 

1st.  Circumscribe  the  circle  with  a  square,  draw  the  diagonals  of 
the  square  P.  0  and  JV.  Q,  and  the  diameters  of  the  circle  Jl.  B 
and  E.  F,  also  through  the  intersections  of  said  diagonals  with  the 
circumference,  draw  the  chords  R.  R,  R.  R,  continued  to  meet 
the  line  G.  L  in  Fand  Y! 

2nd.  Put  the  square  in  perspective  as  before  shewn,  draw  the 
diagonals  JV.  D',  and  0.  D,  and  the  radials  Y.  S  and  Y.f  S. 

3rd.  From  */Z,  draw  Jl.  S,  and  through  the  intersection  of  the  dia- 
gonals draw  E.  F  parallel  to  JV.  0. 

4th.  Through  the  points  of  intersection  thus  found,  viz:  Jl.  B.  E. 
F.  R.  R.  R.  R  trace  the  curve. 

NOTE  1. — This  method  gives  eight  points  through  which  to  trace  the  curve, 
and  as  these  points  are  equidistant  in  the  plan,  it  follows  that  if  the  points 
were  joined  by  right  lines  it  would  give  the  perspective  representation  of  an 
OCTAGON;  by  drawing  diameters  midway  between  those  already  drawn  on 
the  plan,  eight  other  points  in  the  curve  may  be  found.  This  would  give  six- 
teen points  in  the  curve,  and  render  the  operation  of  tracing  much  more  correct. 

NOTE  2. — A  CIRCLE  in  perspective  may  be  considered  as  a  polygon  of  an  in- 
finite number  of  sides,  or  as  a  figure  composed  of  an  infinite  number  of 
points,  and  as  any  point  in  the  curve  may  be  found,  it  follows  that  every 
point  may  be  found,  and  each  be  positively  designated  by  an  intersection; 
in  practice  of  course  this  is  unnecessary,  but  the  student  should  remember,* 
that  the  more  points  he  can  positively  designate  without  confusion,  the  more 
correct  will  be  the  representation. 


PLATE   XLVII. 

LINE    OF    ELEVATION. 


FIGURE  2 


Is  the  plan  of  a  square  whose  side  is  nine  feet,  each  side  is  divided 
into  nine  parts,  and  lines  from  the  divisions  drawn  across  in  oppo- 
site directions;  the  surface  is  therefore  divided  into  eighty-one 
squares.  G.  L,  fig.  1,  is  the  base  line  and  I).  D  the  horizon. 


96  PLATE    XLVII. 

1st. —  To  put  the  plan  with  its  divisions  in  perspective,  one  of  its 
sides  N.  O  to  coincide  with  the  perspective  plane. 

Transfer  the  measures  from  the  side  JV.  0,  fig.  2,  to  JV.  0  on  the 
perspective  plane  fig.  1,  and  put  the  plan  in  perspective  by  the 
methods  before  described. 

2nd. —  To  erect  square  pillars  on  the  squares  N.-  Q.  W,  12  feet 
high  and  one  foot  diameter,  equal  to  the  size  of  one  of  the  squares 
on  the  plan. 

1st.  Erect  indefinite  perpendiculars  from  the  corners  of  the  squares. 

2nd.  On  JV*.  JL  one  of  the  perpendiculars  tliat  touches  the  perspec- 
tive plane  lay  off  the  height  of  the  column  JV*.  M  from  the  ac- 
companying scale,  then  JY.  M  is  A  LINE  OF  HEIGHTS  on  which 
the  true  measures  of  the  heights  of  ail  objects  must  be  set. 

3rd.  Two  lines  drawn  from  the  top  and  bottom  of  an  object  on  the 
line  of  heights  to  the  point  of  sight,  point  of  distance,  or  to  any 
other  point  in  the  horizon,  forms  a  scale  for  determining  similar 
heights  on  any  part  of  the  perspective  plan.  To  avoid  confusion 
they  are  here  drawn  to  the  point  B. 

4th.  Through  M  draw  M.  (7,  parallel  to  N.  0,  and  from  C  draw 
a  line  to  the  point  of  sight  which  determines  the  height  of  the 
side  of  the  column,  and  also  of  the  back  column  erected  on  Q, 
and  through  the  intersection  of  the  line  C.  S  with  the  front  per- 
pendicular, draw  a  horizontal  line  forming  the  top  of  the  front 
side  of  the  column  Q. 

5th.  To  determine  the  height  of  the  pillar  at  W,  1st.  draw  a  hori- 
zontal line  from  its  foot  intersecting  the  proportional  scale  JV*.  B 
in  Y;  2nd.  from  Y  draw  a  vertical  line  intersecting  M.  B  in  X ; 
then  Y.  X  is  the  height  of  the  front  of  the  column  W.  By  the 
same  method  the  height  of  the  column  Q  may  be  determined  as 
shewn  at  R.  T. 

3rd. — To  draw  the  Caps  on  the  Pillars. 

1st.  On  the  line  C.  E  a  continuation  of  the  top  of  the  front,  set  off 
the  amount  of  projection  C.  E,  and  through  E  draw  a  ray  to  the 
point  of  sight. 

2nd.  Through  C  draw  a  diagonal  to  the  point  of  distance,  and 
through  the  point  of  intersection  of  the  diagonal  with  the  ray 


PLATE    XLVII.  97 

last  drawn;  draw  the  horizontal  line  H  forming  the  lower  edge 
of  the  front  of  the  cap. 

3rd.  Through  M  draw  a  diagonal  to  the  opposite  point  of  dis- 
tance, which  determines  the  position  of  the  corners  H  and  K, 
from  H  draw  a  ray  to  the  point  of  sight. 

4th.  Erect  perpendiculars  on  all  the  corners,  lay  off  the  height  of 
the  front,  and  draw  the  top  parallel  with  the  bottom.  A  ray  from 
the  corner  to  the  point  of  sight,  will  complete  the  cap. 

The  other  caps  can  be  drawn  by  similar  means. 

As  a  pillar  is  a  square  column  the  terms  are  here  used  indiscrimi- 
nately. 

4th. —  To  erect  Square  Pyramids  on  O  and  P  of  the  same  height  as 
the  Pillars,  with  a  base  of  four  square  feet,  as  shewn  in  the  plan. 

1st.  Draw  diagonals  to  the  plan  of  the  base,  and  from  their  inter- 
section at  R  draw  the  perpendicular  R.  T1. 

2nd.  From  R  draw  a  line  to  the  proportional  scale  N.  B,  and 
draw  the  vertical  line  Z.  G,  which  is  the  height  of  the  pyramid. 

3rd.  Make  R.  T'  equal  to  Z.  G,  and  from  the  corners  of  the  per- 
spective plan  draw  lines  to  Tf,  which  complete  the  front  pyramid. 

4th.  A  line  drawn  from  T  to  the  point  of  sight  will  determine  the 
height  of  the  pyramid  at  a. 

NOTE  1. — The  point  of  sight  S  shewn  in  front  of  the  column  TF,  must  be 
supposed  to  be  really  a  long  distance  behind  it,  but  as  we  only  see  the  end 
of  a  line  proceeding  from  the  eye  to  the  point  of  sight,  we  can  only  represent 
it  by  a  dot. 

NOTE  2. — A  part  of  the  front  column  has  been  omitted  for  the  purpose  of 
shewing  the  perspective  sections  of  the  remaining  parts,  the  sides  of  these 
sections  are  drawn  toward  the  point  of  sight,  the  front  and  back  lines  are 
horizontal.  The  upper  section  is  a  little  farther  removed  from  the  horizon, 
and  is  consequently  a  little  wider  than  the  lower  section.  This  may  be 
taken  as  an  illustration  of  the  note  to  paragraph  29  on  page  87,  to  which 
the  reader  is  referred. 

NOTE  3. — The  dotted  lines  on  the  plan  shew  the  direction  and  boundaries  of 
the  shadows ;  they  have  been  projected  at  an  angle  of  45°  with  the  plane 
of  the  picture. 


13 


98 


PLATE  XLVIII. 


FIGURE  1. 


To  draw  a  Series  of  Semicircular  Jlrches  viewed  directly  in  front, 
forming  a  Vaulted  Passage,  with  projecting  ribs  at  intervals,  as 
shewn  by  the  tinted  plan  below  the  ground  line. 

1st.  From  the  top  of  the  side  walls  JY.  I  and  0.  K,  draw  the  front 
arch  from  the  centre  H,  and  radiate  the  joints  to  its  centre. 

2nd.  From  the  centre  H  and  the  springing  lines  of  the  arch,  and 
from  the  corners  Ji  and  M  draw  rays  to  the  point  of  sight. 

3rd.  From  Jl  and  M  set  off  the  projection  of  the  ribs,  and  draw 
rays 'from  the  points  so  set  off  to  the  point  of  sight. 

4th.  Transfer  the  measurements  of  Jf'.  B".  C/f,  &c.,  on  the  plan,to 
Jl .  B .  C",  &,c.,  on  the  ground  line,  and  from  them  draw  diago- 
nals to  the  point  of  distance,  intersecting  the  r.ay  Ji.  S  in  B.  C. 
D,  &c. 

5th.  From  the  points  of  intersection  in  Jl.  S  draw  lines  parallel  to 
the  base  line  to  intersect  M.  S.  This  gives  the  perspective  plans 
of  the  ribs. 

6th.  Erect  perpendiculars  from  the  corners  of  the  plans  to  inter- 
sect the  springing  lines,  and  through  these  intersections  draw 
horizontal  dotted  lines,  then  the  points  in  which  the  dotted  lines 
intersect  the  ray  drawn  from  H  the  centre  of  the  front  arch,  will  | 
be  the  centres  for  drawing  the  other  arches;  R  being  the  centre 
for  describing  the  front  of  the  first  rib. 

7th.  The  joints  in  the  fronts  of  the  projecting  ribs  radiate  to  their 
respective  centres,  and  the  joints  in  the  soffit  of  the  arch  radiate 
to  the  point  of  sight. 

NOTE. — No  attempt  is  made  in  this  diagram  to  project  the  shadows,  as  it 
would  render  the  lines  too  obscure.  But  the  front  of  each  projection  is  tint- 
ed to  make  it  more  conspicuous. 


OP  TBX 

•UN  17  BE  SIT  7] 


ARCHES  IN  PERSPECTIVE. 


ARCHES  WPERSPEt  TfVE. 


-~:-..,  s 


Fia.2. 


A         B 


F          G 


PLATE  XLVIII.  99 

FIGURE  2. 

To  draw  Semicircular  or  Pointed  Jlrcades  on  either  side  of  the 
spectator,  running  parallel  to  the  middle  visual  ray.  N.  P  and 
Q,  O  the  width  of  the  arches  being  given,  and  P.  Q  the  space  be- 
tiveen  them. 

1st.  From  N.  P.  Q  and  0  erect  perpendiculars,  make  them  all  of 
equal  length,  and  draw  E.  F  and  M.  J. 

2nd.  FOR  THE  SEMICIRCULAR  ARCHES,  bisect  E.  F  in  C,  and 
from  E.  C.  F.  0  and  Q,  draw  rays  to  the  point  of  sight. 

3rd.  From  C,  describe  the  semicircle  E.  F. 

4th.  Let  the  arches  be  the  same  distance  apart  as  the  width  Q.  0, 
then  from  0  draw  a  diagonal  to  the  point  of  distance,  cutting  Q. 
S  in  R,  from  R  draw  a  diagonal  to  the  opposite  point  of  distance 
cut'.nig  0.  S  in  V,  from  V  draw  a  diagonal  to  D,  cutting  Q.  S  in 
W,  and  from  W  to  Z>',  cutting  0.  S  in  X. 

5ih.  Through  R.  V.  £Fand  X,  draw  horizontal  lines  to  intersect 
the  rays  0.  S  and  Q.  S,  and  on  the  intersections  erect  perpen- 
diculars to  meet  the  rays  drawn  from  E  and  F. 

6th.  Connect  the  tops  of  the  perpendiculars  by  horizontal  lines, 
and  from  their  intersections  with  the  ray  drawn  from  C  in  1,  2, 
3  and  4,  describe  the  retiring  arches. 

7th.  FOR  THE  GOTHIC  ARCHES,  (let  them  be  drawn  the  same  dis- 
tance apart  as  the  semicircular,)  continue  the  horizontal  lines 
across  from  R  and  F,  to  intersect  the  rays  P.  S  and  N.  S,  and 
from  the  points  of  intersection  erect  perpendiculars  to  intersect 
the  rays  drawn  from  M  and  J. 

8th.  From  M  and  J  successively,  with  a  radius  M.  J,  describe  the 
front  arch,  and  from  .fiTthe  crown^draw  a  ray  to  S;  from  Ji  and 
B  with  the  radius  Ji.  B,  describe  the  second  arch,  and  from  K 
and  L,  describe  the  third  arch. 

NOTE  . — All  the  arches  in  this  plate  are  parallel  to  the  plane  of  the  picture, 
and  although  each  succeeding  arch  is  smaller  than  the  arch  in  front  of  it,  all 
may  be  described  with  the  compasses. 


100 


PLATE    XLIX. 

TO  DESCRIBE  ARCHES  ON  A  VANISHING  PLANE. 


FIGURE  1. 

The  Front  JLrch  A.  N.  B,  the,  Base  Line  G.  L,  Horizon  D.  S, 
Point  of  Sight  S,  arid  Point  of  Distance  D,  being  given. 

1st.  Draw  H.  J  across  the  springing  line  of  the  arch,  and  construct 

the  parallelogram  E.  F.  J.  H. 
2nd.  Draw  the  diagonals  H.  F  and  J.  E,  and  a  horizontal  line  K. 

My  through  the  points  where  the  diagonals   intersect  the  given 

arch.    Then  H.  K.  N.  M  and  J,  are,  points  in  the  curve  which  are 

required  to  be  found  in  each  of  the  lateral  arches. 
3rd.  From  F  and  B,  drawT  rays  to  the  point  of  sight  S.     Then  if 

we  suppose  the  space  formed  by  the  triangle  B.  S.  F  to  be  a 

plane  surface,  it  will  represent  the  vanishing  plane  on  which  the 

arches  are  to  be  drawn. 
4th.  From  B,  set  off  the  distance  B.  Jl  to  Z,  and  draw  rays  from 

Z.  J  and  (7,  to  the  point  of  sight. 
5th.  From  Z,  draw  a  diagonal  to  the  point  of  distance,  cutting  B. 

Sin  0;  through  0,  draw  a  horizontal  line  cutting  Z.  S  in  P; 

from  P,  draw  a  diagonal  intersecting  B.  S  in   Q;  through   Q, 

draw  a  horizontal  line,  cutting  Z.  S  in  /?,  and  so  on  for  as  many 

arches  as  may  be  required. 
6th.   From  0.  Q.  S.    U,  erect  perpendiculars,  cutting  F.  S  in  V. 

W.  X.  Y. 
7th.  Draw  the  diagonals  /.  V^  F.  7,  &c.  as  shewn  in  the  diagram, 

and  from  their  intersection  erect  perpendiculars   to  meet  F.  S; 

through  which  point  and  the  intersections  of  the  diagonals  with 

C.  S  trace  the  curves. 


FIGURE     2. 


To  draw  Receding  Jlrches  on  the  Vanishing  Plane  J.  S.  D,  with 
Piers  between  them.,  corresponding  with  the  given  front  view,  the 
Piers  to  have  a  Square  Base  with  a  side  equal  to  C.  D. 

I  1st.  From  D  on  the  base  line,  set  off  the  distances  D.  C,  C.  B 


Ill  - 


THE  OJiJECT  AX/>  POINT  OF  VIEW  GJl'KX 
TO  F1XJ)  THE  PKItxrKCTrVK  7>LAW 


.LY/J  VANISHING  IVUXTS. 


UNIVERSITY 


PLATE    XLIX.  101 

and  B.  A  to  D.  E,  E.  F  and  F.  G,  and  from  E.  F.  G,  £,c.  draw 

diagonals  to  the  point  of  distance  to  intersect  D.  S. 
2nd.  From  the  intersections  in  D.  S,  erect  perpendiculars;  draw 

the  parallelogram  M.  JY.  H.  I  around  the  given  front  arch,  the 

diagonals  M.  I  and  H.  N,  and  the  horizontal  line  L.  K,  prolong 

H.  /to  JandJlf.  JVto  V. 
3rd.  From  B.  C.  D.  M.  f.  J.  K  and  V^  draw  rays  to  the  point  of 

sight,  put  the  parallelograms  and  diagonals  in  perspective  at  0.  P 

V.  W  and  at  Q.  W.  R.  X,  and  draw  the  curves  through  the  points 

as  in  the  last  diagram. 
4th.  From  i  where  E.  D1  cuts  D.  S,  draw  a  horizontal  line  cutting 

B.  S  in  h,  and  from  h  erect  a  perpendicular  cutting  M.  S  in  k. 
5th.  From  F,  the  centre  of  the  front  arch,  draw  a  ray  to  the  point 

of  sight,  and  from  k,  draw  a  horizontal  line  intersecting  it  in  Z. 

Then  Z  is  the  centre  for  describing  the  back  line  of  the  arch  with 

the  distance  Z.  k  for  a  radius. 
NOTE. — The  backs  of  the  side  arches  are  found  by  the  same  method  as  the 

front?  of  those  arches.     The  lines  are  omitted  to  avoid  confusion. 
The  projecting  cap  in  this  diagram  is  constructed  in  the  same  manner  as  the 

caps  of  the  pillars  in  Plate  47. 


PLATE    L. 

APPLICATION    OF   THE  CIRCLE  WHEN    PARALLEL   TO   THE 
PLANE  OF  THE  PICTURE. 


V.  L  is  the  horizon,  and  S  the  point  of  sight. 
FIGURE  1 


To  draw  a  Semicircular  Thin  Band  placed  abov?  the  horizon. 


Let  the  semicircle  Jl.  B  represent  the  front  edge  <)f:t? 

B  the  diameter,  and  C  the  centre. 
1st.  From  JL.  Cand  B,  draw  rays  to  the  point  of  sight. 
2nd.  From  C  the  centre,  lay  off  toward  B,  the  breadth  of  the  band 

C.  E. 
3rd.  From  E,  draw  a  diagonal  to  the  point  of  distance,  intersect- 

ing C.  S  in  F.     Then  F  is  the  centre  for  describing  the  back  of 

the  band. 


102  PLATE    L. 

4th.  Through  F,  draw  a  horizontal  line  intersecting  ./I.  S  in  K. 
and  B.  S  in  L.  Then  F.  K  or  F.  L  is  the  radius  for  describing 
the  back  of  the  band. 

FIGURE  2. 


To  draw  a   Circular  Hoop  with  its  side  resting  on  the  Horizon. 

The  front  circle  Jl.  H.  B.  K,  diameter  JL.  B,  and  centre  C  being 

given. 

1st.  From  A.  C  and  B,  draw  rays  to  the  point  of  sight. 
2nd.  From  C  the  centre,  lay  off  the  breadth  of  the  hoop  at  E. 
3rd.  From  E,  draw  a  diagonal  to   D',  intersecting   C.  Sin  F,  and 

through  F,  draw  a  horizontal  line  intersecting  Jl.  S  in  K,  and  B 

SmL. 
4th.  From  F  with  a  radius  F.  L  or  F.  K,  describe  the  back  of  the 

curve. 

FIGURE  3. 


To  draw  a  Cylindrical  Tub  placed  below  the  Horizon,  whose  dia- 
meter,  depth  and  thickness  are  given. 

1st.  From  the  centre  C  describe  the  concentric  circles  forming  the 

thickness  of  the  tub,  lay  off  the  staves  and  radiate  them  toward 

C. 
2nd.   Proceed  as  in  figs.  1  and  2  to  draw  rays  and  a  diagonal  to 

find  the  point  F,  and  from  F  describe  the  back  circles  as  before ; 

the  hoop  may  be  drawn  from  F,  by  extending  the  compasses  a 

little. 
3rd.  Radiate  all  the  lines  that  form  the  joints  on  the  sides  of  the 

tub  toward  the  point  of  sight. 


FIGURE  4 


Is  a  hollow  cylinder  placed  below  the  horizon,  and  must  be  drawn 
by  the  same  method  as  the  preceding  figures ;  the  letters  of  re- 
ference are  the  same. 

NOTE. — The  objects  in  this  Plate  are  tinted  to  shew  the   different  surface 
more  distinctly  without  attempting  to  project  the  shadows. 


103 


PLATE    LI. 


The  object  and  point  of  view  given,  to  find  the  Perspective  Plane 
and  Vanishing  Points. 

Rule  1. — The  PERSPECTIVE  PLANE  must  be  drawn  perpendicular 
to  the  middle  visual  ray. 

Rule  2. — The  VANISHING  POINT  of  a  line  or  plane  is  found  by 
drawing  a  line  through  the  station  point  parallel  with  such  line 
or  plane  to  intersect  the  perspective  plane.  The  point  in  the  hori- 
zon immediately  over  the  intersection  so  found,  is  the  vanishing 
point  of  all  horizontal  lines  in  said  plane,  or  on  any  plane  parallel 
to  it. 

1st.  Let  the  parallelogram  E.  F.  G.  Hbe  the  plan  of  an  object  to 
be  put  in  perspective,  and  let  Q  be  the  position  of  the  spectator 
viewing  it,  (called  the  point  of  view  or  station  point,)  with  the 
eye  directed  toward  K,  then  Q.  K  will  be  the  central  visual  ray, 
and  K  the  point  of  sight.  Draw  F.  Q  and  H.  Q,  these  are  the 
external  visual  rays. 

NOTE. — The  student  should  refer  to  paragraphs  30  and  31,  page  87,  for  the 
definitions  of  station  point  and  point  of  sight. 

2nd.  Draw  P.  0  at  right  angles  to  Q.  K,  touching  the  corner  of 
the  given  object  at  E,  then  P.  O  will  be  the  base  of  the  perspec- 
tive plane. 

NOTE. — This  position  of  the  perspective  plane,  is  the  farthest  point  from  the 
spectator  at  which  it  can  be  placed,  as  the  whole  of  the  object  viewed  must 
be  behind  it;  but  it  may  be  placed  at  any  intermediate  point  nearer  the 
spectator  parallel  with  P.  0. 

3rd.  Through  Q  draw  Q.  P  parallel  with  E.  F,  intersecting  the 
perspective  plane  in  P,  then  P  is  the  vanishing  point  of  the  lines 
E.  F  and  G.  H. 

4th.  Through  Q  draw  Q.  0,  parallel  to  E.  H,  intersecting  the 
perspective  plane  in  0,  then  O  is  the  vanishing  point  for  E.  H 
and  F.  G. 

5th.  If  we  suppose  the  station  point  to  be  removed  to  Jl,  then  A. 
M  will  be  the  central  visual  ray,  Ji.  F  and  Jl.  H  the  external 
rays,  and  B.  D  the  perspective  plane,  B  the  vanishing  point  of 


104  PLATE    LI. 

E.  F  and  G.  H,  and  the  vanishing  point  of  E.  H  and  F.  G  will 
be  outside  the  plate  about  ten  inches  distant  from  JL*  in  the  direc- 
tion of  A.  C. 

6th.  If  the  station  point  be  removed  to  K,  it  will  be  perceived  that 

E.  H  and  F.  G  will  have  no  vanishing  point,  because  they  are 

perpendicular  to  the  middle  ray,  and  a  line  drawn  through  the 

station  point  parallel  with  the  side  E.  H  will  also  be  parallel  with 

the  perspective  plane,  consequently  could  never  intersect  it. 

7th.  The  sides   E.  F  and  G.  H  of  the  plan,  would  vanish  in  the 

point  of  sight,  but  if  an  elevation  be  drawn  on  the  plan  in   that 

position  which  should  extend  above  the  horizon,  then  neither  of 

those  sides  could  be  seen,  and  the  drawing  would  very  nearly 

approach  to  a  geometrical  elevation  of  the  same  object. 

NOTE. — In  the  explanation  of  this  plate,  the  intersections  giving  the  point  of 

sight  and  vanishing  points,  are  made  in  the  perspective,  plane,  which  the 

student  will  remember  when  used  in  this  connection,  is  equivalent  to  the  base 

line  or  ground  line  of  the  picture,  being  the  seat  or  position  of  the  plane  on 

which  the  drawing  is  to  be  made;  but  we  must  suppose  these  points  to  be 

elevated  to  the  height  of  the  eye  of  the  spectator;  in  practice,  these  points 

must  be  set  off  on  the  horizontal  line  as  described  in  paragraph  32,  page  87. 


PLATE  LII. 


To  delineate  the  perspective  appearance  of  a   Cube  viewed  acci- 
dentally and  situated  beyond  the  Perspective  Plane. 


FIGURE   L 

Let  Jl.  B.  C.  D  be  the  plan  of  the  cube,  S  the  station  point,  S.  T 
the  middle  visual  ray  and  B.  L  the  base  line,  or  perspective 
plane. 

1st.  Continue  the  sides  of  the  plan  to  the  perspective  plane  as 
shewn  by  the  dotted  lines,  intersecting  it  in  M.  E.  JVand  0. 

2nd.  From  the  corners  of  the  plan  draw  rays  to  the  station  point, 
intersecting  the  perspective  plane  in  a.  d.  b.  c. 

3rd.  Through  S,  draw  S.  F  parallel  to  Jl.  D,  and  S.  G  parallel 
to  D.  C.  Then  F  is  the  vanishing  point  for  the  sides  A.  D  and 
B.  C;  and  G  is  the  vanishing  point  for  the  sides  A.  B  and  D,  C. 


OBJECT  INCLINED  TO  THE  PLANE  OF  DELINEATION. 


W 'rl- Minifie . 


PLAN  AND  PERSPECTIVE  VIEW: 

Fia  7 


<^1P^ 

[BIIVBRSITT] 


PLATE    LIT.  105 

FIGURE  2. 

4th.  Transfer  these  intersections  from  B.  £,  fig.  1,  to  B.  L,  fig.  2, 
and  the  vanishing  points  F  and  G  to  the  horizon,  as  shewn  by 
the  dotted  lines. 

5th.  From  E  and  M,  draw  lines  to  the  vanishing  point  G,  and 
from  JY  and  0,  draw  lines  to  the  vanishing  point  F.  Then  the 
trapezium  A.  B.  C.  D  formed  by  the  intersection  of  these  lines,  is 
the  perspective  view  of  the  plan  of  the  cube. 

6th.  To  DRAW  THE  ELEVATION.  At  M.  E.  JV*  and  0  erect  per- 
pendiculars and  make  them  equal  to  the  side  of  the  cube. 

7th.  From  the  tops  of  these  perpendiculars  draw  lines  to  the  op- 
posite vanishing  points  as  shewn  by  the  dotted  lines,  their  inter- 
section will  form  another  trapezium  parallel  to  the  first,  repre- 
senting the  top  of  the  cube. 

8th.  From  JL.  D  and  C,  erect  perpendiculars  to  complete  the  cube. 

NOTE. — It  is  not  necessary  to  erect  perpendiculars  from  all  the  points  of  in- 
tersection, to  draw  the  representation,  but  it  is  done  here  to  prove  that  the 
height  of  an  object  may  be  set  on  any  perpendicular  erected  at  the  point 
where  the  plane, or  line,  or  a  continuation  of  a  line  intersects  the  perspective 
plane;  one  such  line  of  elevation  is  generally  sufficient. 

9th.  To  draw  the  figure  with  one  line  of  heights,  proceed  as  fol- 
lows: from  A.  D  and  C,  erect  indefinite  perpendiculars. 

10th.  Make  E.  H equal  to  the  side  of  the  cube,  and  from  //"draw 
a  line  to  G,  cutting  the  perpendiculars  from  D  and  C  in  K  and  L. 

1 1th.  From  K,  draw  a  line  to  F,  cutting  Jl.  P  in  P ;  from  L,  draw 
a  line  to  F,  and  from  P,  draw  a  line  to  G,  which  completes  the 
figure. 

NOTE.- — The  student  should  observe  how  the  lines  and  horizontal  planes  be- 
come diminished  as  they  approach  toward  the  horizon,  each  successive  line 
becoming  shorter,  and  each  plane  narrower  until  at  the  height  of  the  eye,  the 
whole  of  the  top  would  be  represented  by  a  straight  line.  I  would  here  re- 
mark, that  it  would  very  materially  aid  the  student  in  his  knowledge  of  per- 
spective, if  he  would  always  make  it  a  rule  to  analyze  the  parts  of  every  dia- 
gram he  draws,  observe  the  changes  which  take  place  in  the  forms  of  ob- 
jects when  placed  in  different  positions  on  the  plan,  and  when  they  are 
placed  above  or  below  the  horizon  at  different  distances;  this  'would  enable 
him  at  once  to  detect  a  false  line,  and  would  also  enable  him  to  sketch  from 
nature  with  accuracy.  Practice  this  always  until  it  becomes  a  HABIT,  and  I 
can  assure  you  it  will  be  a  source  of  much  gratification. 


14 


106 


PLATE  LIII. 

TO   DRAW   THE   PERSPECTIVE   VIEW   OF   A  ONE   STORY 
COTTAGE,   SEEN   ACCIDENTALLY. 


FIGURE  1. 

Let  Jl.  B.  C.  D  be  the  plan  of  the  cottage,  twenty  feet  by  four- 
teen feet,  drawn  to  the  accompanying  scale;  the  shaded  parts 
shew  the  thickness  of  the  walls  and  position  of  the  openings, 
the  dotted  lines  outside  parallel  with  the  walls,  give  the  projec- 
tion of  the  roof,  and  the  square  E.  F.  G.  H,  the  plan  of  the 
chimney  above  the  roof. 

Let  P.  L  be  the  perspective  plane  and  S  the  station  point. 

1st.  Continue  the  side  B.  D  to  intersect  the  perspective  plane  in 
H,  to  find  the  position  for  a  line  of  heights. 

2nd.  From  all  the  corners  and  jambs  on  the  plan,  draw  rays  toward 
the  station  point  to  intersect  the  perspective  plane. 

3rd.  Through  S  draw  a  line  parallel  to  the  side  of  the  cottage  D. 
C,  to  intersect  the  perspective  plane  in  L.  This  gives  the  vanish- 
ing point  for  the  ends  of  the  building  and  for  all  planes  parallel 
to  it,  viz :  the  side  of  the  chimney,  and  jambs  of  the  door  and 
windows. 

4th.  Through  S  draw  a  line  parallel  with  B.  D,  to  intersect  the 
perspective  plane,  which  it  would  do  at  some  distance  outside  of 
the  plate;  this  intersection  would  be  the  vanishing  point  for  the 
sides  of  the  cottage,  for  the  tops  and  bottoms  of  the  windows, 
the  ridge  and  eaves  of  the  roof,  and  for  the  front  of  the  chimney. 

****MWMWWfW!MMM*IVM« 

FIGURE  2. 


Let  us  suppose  the  parallelogram  P.  L.  W.  X  to  be  a  separate  piece 
of  paper  laid  oil  the  other,  its  top  edge  coinciding  with  the  per- 
spective plane  of  fig.  1,  and  its  bottom  edge  W.  X  to  be  the  base 
of  the  picture,  then  proceed  as  follows  : 

1st.  Draw  the  horizontal  line  R.  T  parallel  to  W.  X  and  five  feet 
above  it. 


PLATE    LIII.  107 

2nd.  Draw  H.  K  perpendicular  to  P.  L  for  a  line  of  heights. 

3rd.  Draw  a  line  from  K  to  the  vanishing  point  without  the  pic- 
ture, which  we  will  call  Z ;  this  will  represent  the  line  H.  B  of 
fig.  1,  continued  indefinitely. 

4th.  From  b  and  d  draw  perpendiculars  to  intersect  the  last  line 
drawn,  in  o  and  e,  which  will  determine  the  perspective  length  of 
the  front  of  the  house. 

5th.  On  K.  H  set  off  twelve  feet  the  height  of  the  walls,  at  0,  and 
from  0  draw  a  line  to  the  vanishing  point  Z,  intersecting  d.  e  in 
m  and  b.  o  in  n. 

6th.  From  m  and  e  draw  vanishing  lines  to  T,  and  a  perpendicu- 
lar from  c  intersecting  them  in  Fand  s ;  this  will  give  the  cor- 
ner F,  and  determine  the  depth  of  the  building. 

7th.  Find  the  centre  of  the  vanishing  plane  representing  the  end, 
by  drawing  the  diagonals  m.  Y  and  e.  s,  and  through  their  inter- 
section draw  an  indefinite  perpendicular  u.  v,  which  will  give  the 
position  of  the  gable. 

8th.    To    FIND    THE    HEIGHT    OF    THE    GABLE,  Set  off  its    proposed 

height,  say  7'  0",  from  0  to  JVon  the  line  of  heights,  from  JV  draw 
a  ray  to  Z,  intersecting  e.  d  in  W,  and  from  W  draw  a  vanishing 
line  to  T  intersecting  u.  v  in  v,  then  v  is  the  peak  of  the  gable. 

9ih.  Join  m.  v,  and  prolong  it  to  meet  a  perpendicular  drawn 
through  the  vanishing  point  T,  which  it  will  do  in  V,then  N  is  the 
vanishing  point  for  the  inclined  lines  of  the  ends  of  the  front  half 
of  the  roof.  The  ends  of  the  backs  of  the  gables  will  vanish  in  a 
point  perpendicularly  below  F,  as  much  below  the  horizon  as  V 
is  above  it. 

10th.  FOR  THE  ROOF.  Through  v  draw  v.  y  to  Z  without,  to 
form  the  ridge  of  the  roof,  from  /  let  fall  a  perpendicular  to  inter- 
sect y.  v  in  w,  through  w  draw  a  line  to  the  vanishing  point  V  to 
form  the  edge  of  the  roof.  From  d  let  fall  a  perpendicular  to  in- 
tersect V.  w,  and  from  the  point  of  intersection  draw  a  line  to  Z  to 
form  the  front  edge  of  the  roof,  from  a  let  fall  a  perpendicular  to  de- 
fine the  corner  x,  and  from  x  draw  a  line  to  V  intersecting  w.  y  in 
y,  which  completes  the  front  half  of  the  roof;  from  w  draw  a  line 
to  the  vanishing  point  below  the  horizon,  from  c  let  fall  a  perpen- 
dicular to  intersect  it  in  g,  and  through  g  draw  a  line  to  Z,  which 
completes  the  roof. 

llth.  FOR  THE  CHIMNEY.  Set  off  its  height  above  the  ridge  at 
JV/,  from  M  draw  a  line  toward  the  vanishing  point  Z,  intersect- 
ing o.  b  in  U,  from  U  draw  a  line  to  the  vanishing  point  J1,  which 


108  PLATE  LIII. 

gives  the  height  of  the  chimney,  bring  down  perpendiculars  from 
rays  drawn  from  G.  F  and  E,  fig.  1,  and  complete  the  chimney 
by  vanishing  lines  drawn  for  the  front  toward  Z  and  for  the  side 
toward  T. 

12th  FOR  THE  DOOR  AND  WINDOWS.  Set  off  their  heights  at 
P.  Q  and  draw  lines  toward  Z,  bring  down  perpendiculars 
from  the  rays  as  before,  to  intersect  the  lines  drawn  toward  Z; 
these  lines  will  determine  the  breadth  of  the  openings.  The 
breadth  of  the  jambs  are  found  by  letting  fall  perpendiculars  from 
the  points  of  intersection,  the  top  and  bottom  lines  of  the  jambs 
are  drawn  toward  T. 

NOTE  1. — As  the  bottom  of  the  front  fence  if  continued,  would  intersect  the 
base  line  at  K  the  foot  of  the  line  of  heights,  and  its  top  is  in  the  horizon,  it 
is  therefore  five  feet  high. 

NOTE  2. — The  whole  of  the  lines  in  this  diagram  have  been  projected  accord- 
ing to  the  rules,  to  explain  to  the  learner  the  methods  of  doing  so,  and  it  will 
be  necessary  for  him  to  do  so  until  he  is  perfectly  familiar  with  the  subject. 
But  if  he  will  follow  the  rule  laid  down  at  the  end  of  the  description  of  the  last 
plate,  he  will  soon  be  enabled  to  complete  his  drawing  by  hand,  after  pro- 
jecting the  principal  lines,  but  it  should  not  be  attempted  too  early,  as  it 
will  beget  a  careless  method  of  drawing,  and  prevent  him  from  acquiring  a 
correct  judgment  of  proportions. 


PLATE    LIV. 


FRONTISPIECE 

Is  a  perspective  view  of  a  street  60  feet  wide,  as  viewed  by  a  per- 
son standing  in  the  middle  of  the  street  at  a  distance  of  134  feet 
from  the  perspective  plane,  and  at  an  elevation  of  20  feet  from 
the  ground  to  the  height  of  the  eye.  The  horizon  is  placed  high 
for  the  purpose  of  shewing  the  roofs  of  the  two  story  dwellings. 
The  dimensions  of  the  different  parts  are  as  follows  : 

1st. — DISTANCES  ACROSS  THE  PICTURE. 

Centre  street  between  the  houses  60  0  feet  wide. 

Side  walks,  each  10  0       " 

Middle  space  between  the  lines  of  railway     46       " 
Width  between  the  rails  49       " 


PLATE    LIV.  109 

Depth  of  three  story  warehouse  40  feet. 

Depth  of  yard  in  the  rear  of  warehouse         20    " 
Depth  of  two  story  dwelling  on  the  right     30    " 

DISTANCES  FROM  THE  SPECTATOR,  IN  THE  LINE  OF  THE 
MIDDLE  VISUAL  RAY. 

From  spectator  to  plane  of  the  picture  134  feet. 

From  plane  of  picture  to  the  corner  of  buildings  50  " 

Front  of  each  house  20  " 

Front  of  block  of  7  houses  20  feet  each  140  " 

Breadth  of  street  running  across  between  the  blocks  60  " 

Depth  of  second  block  same  as  the  first  140  " 

Depth  of  houses  on  the  left  of  the  picture,  behind  >  .Q  (( 
the  three  story  warehouses                                     3 

To  DRAW  THE  PICTURE. 

1st.  Let  Cbe  the  centre  of  the  perspective  plane,  H.  L  the  hori- 
zon, S  the  point  of  sight. 

2nd.  From  C  on  the  line  P.  P,  lay  off  the  breadth  of  the  street 
thirty  feet  on  each  side,  at  0  and  60,  making  sixty  feet,  and  from 
those  points  draw  rays  to  the  point  of  sight;  these  give  the  lines 
of  the  fronts  of  the  houses. 

3rd.  From  0  lay  off  a  point  50  feet  on  P.  Py  and  draw  a  diagonal 
from  that  point  to  the  point  of  distance  without  the  picture ;  the 
intersection  of  that  diagonal  with  the  ray  from  0,  determines  the 
corner  of  the  building;  from  the  point  of  intersection  erect  a  per- 
pendicular to  B. 

4th.  From  50,  lay  off  spaces  of  20  feet  each  at  70,  90  and  so  on, 
and  from  the  points  so  laid  off  draw  diagonals  to  determine  by 
their  intersection  with  the  ray  from  0,  the  depth  of  each  house. 

5th.  After  the  depth  on  0.  S  is  found  for  three  houses,  the  depths 
of  the  others  may  be  found  by  drawing  diagonals  to  the  oppo- 
site point  of  distance  to  intersect  the  ray  60  S,  as  shewn  by  the 
dotted  lines. 

NOTE. — As  a  diagonal  drawn  to  the  point  of  distance  forms  an  angle  of  45° 
with  the  plane  of  the  picture,  it  follows  that  a  diagonal  drawn  from  a  ray  to 
another  parallel  ray,  will  intercept  on  that  ray  a  space  equal  to  the  distance 
between  them.  Therefore  as  the  street  in  the  diagram  is  60  feet  wide  and 
the  front  of  each  house  is  20  feet,  it  follows  that  a  diagonal  drawn  from  one 
side  of  the  street  to  the  other  will  intercept  a  space  equal  to  the  fronts  of 
three  houses,  as  shewn  in  the  drawing. 

6th.  Lay  off  the  dimensions  on    the  perspective    plane,  of  the 


HO  PLATE    LIV. 

depth  of  the  houses,  and  the  position  of  the  openings  on  the 
side  of  the  warehouse,  and  draw  rays  to  the  point  of  sight  as 
shewn  by  the  dotted  lines. 

7th.  At  0  erect  a  perpendicular  to  D  for  a  line  of  heights;  on  this 
line  all  the  heights  must  be  laid  off  to  the  same  scale  as  the  mea- 
sures on  the  perspective  plane,  and  from  the  points  so  marked 
draw  rays  to  the  point  of  sight  to  intersect  the  corner  of  the 
building  at  B.  For  example,  the  height  of  the  gable  of  the  ware- 
house is  marked  at  ^,  from  Jl  draw  a  ray  toward  the  point  of 
sight  intersecting  the  corner  perpendicular  at  B ;  then  from  jB, 
draw  a  horizontal  line  to  the  peak  of  the  gable ;  the  dotted  lines 
shew  the  position  of  the  other  heights. 

8th.  To  find  the  position  of  the  peaks  of  the  gables  on  the  houses 
in  the  rear  of  the  warehouses,  draw  rays  from  the  top  and  bottom 
corner  of  the  front  wall  to  the  point  of  sight,  draw  the  diagonals 
as  shewn  by  the  dotted  lines,  and  from  their  intersection  erect  a 
perpendicular,  which  gives  the  position  of  the  peak,  the  intersec- 
tion of  diagonals  in  this  manner  will  always  determine  the  perspec- 
tive centre  of  a  vanishing  plane.  The  height  may  be  laid  off  on 
0.  D  at  Z),  and  a  ray  drawn  to  the  point  of  sight  intersecting  the 
corner  perpendicular  at  (7,  then  a  parallel  be  drawn  from  C  to 
intersect  a  perpendicular  from  the  front  corner  of  the  building 
at  £",  and  from  that  intersection  draw  a  ray  to  the  point  of  sight. 
The  intersection  of  this  ray,  with  the  indefinite  perpendicular 
erected  from  the  intersection  of  the  diagonals,  will  determine  the 
perspective  height  of  the  peak. 

9th.  The  front  edges  of  the  gables  will  vanish  in  a  point  perpen- 
dicularly above  the  point  of  sight,  and  the  back  edges  in  a  point 
perpendicularly  below  it  and  equidistant. 

10th.  As  all  the  planes  shewn  in  this  picture  except  those  parallel 
with  the  plane  of  the  picture  are  parallel  to  the  middle  visual 
ray,  all  horizontal  lines  on  any  of  them  must  vanish  in  the  point 
of  sight,  and  inclined  lines  in  a  perpendicular  above  or  below  it, 
as  shewn  by  the  gables. 


Ill 


SHADOWS, 


1st.  The  quantity  of  light  reflected  from  the  surface  of  an  object, 
enables  us  to  judge  of  its  distance,  and  also  of  its  form  and  posi- 
tion. 

2nd.  On  referring  to  paragraph  9,  page  82,  it  will  be  found  that 
light  is  generally  considered  in  three  degrees,  viz :  light,  shade  and 
shadow;  the  parts  exposed  to  the  direct  rays  being  in  light,  the 
parts  inclined  from  the  direct  rays  are  said  to  be  in  shade,  and 
objects  are  said  to  be  in  shadow,  when  the  direct  rays  of  light  are 
intercepted  by  some  opaque  substance  being  interposed  between 
the  source  of  light  and  the  object. 

3rd.  THE  FORM  OF  THE  SHADOW  depends  on  the  form  and  posi- 
tion of  the  object  from  which  it  is  cast,  modified  by  the  form  and 
position  of  the  surface' on  which  it  is  projected.  For  example,  if 
the  shadow  of  a  cone  be  projected  by  rays  perpendicular  to  its 
axis,  on  a  plane  parallel  to  its  axis,  the  boundaries  of  the  shadow 
will  be  a  triangle;  if  the  cone  be  turned  so  that  its  axis  would 
be  parallel  with  the  ray,  its  shadow  would  be  a  circle;  if  the  cone 
be  retained  in  its  position,  and  the  plane  on  which  it  is  projected 
be  inclined  in  either  direction,  the  shadow  will  be  an  ellipsis, 
the  greater  the  obliquity  of  the  plane  of  projection,  the  more 
elongated  will  be  the  transverse  axis  of  the  ellipsis. 

4th.  SHADOWS  OF  THE  SAME  FORM  MAY  BE  CAST  BY  DIFFER- 
ENT FIGURES:  for  example,  a  sphere  and  a  flat  circular  disk 
would  each  project  a  circle  on  a  plane  perpendicular  to  the  rays 
of  light,  so  also  would  a  cone  and  a  cylinder  with  their  axes  par- 
allel to  the  rays.  The  sphere  would  cast  the  same  shadow  if 
turned  in  any  direction,  but  the  flat  disk  if  placed  edgeways  to 
the  rays,  would  project  a  straight  line,  whose  length  would  be 
equal  to  the  diameter  of  the  disk  and  its  breadth  equal  to  the 
thickness;  the  shadow  of  the  cone  if  placed  sideways  to  the  rays 
would  be  a  triangle,  and  of  the  cylinder  would  be  a  parallelogram. 

5th.  Shadows  of  regular  figures  if  projected  on  a  plane,  retain 
in  some  degree  the  outline  of  the  object  casting  them,  more  or 


112 

less  distorted,  according  to  the  position  of  the  plane;  but  if  cast 
upon  a  broken  or  rough  surface  the  shadow  will  be  irregular. 

6th.  Shadows  projected  from  angular  objects  are  generally  strong- 
ly defined,  and  the  shading  of  such  objects  is  strongly  contrasted; 
thus  if  you  refer  to  the  cottage  on  Plate  53,  you  will  perceive 
that  the  vertical  walls  of  the  front  and  chimney  are  in  light,  fully 
exposed  to  the  direct  rays  of  the  sun,  while  the  end  of  the  cot- 
tage and  side  of  the  chimney  are  in  shade,  being  turned  away 
from  the  direct  rays,  the  plane  of  the  roof  is  not  so  bright  as  the 
vertical  walls,  because, although  it  is  exposed  to  the  direct  rays  of 
light  it  reflects  them  at  a  different  angle,  the  shadow  of  the  pro- 
jecting eaves  of  the  roof  on  the  vertical  wall  forms  a  dark  un- 
broken line,  the  edge  of  the  roof  being  straight  and  the  surface 
of  the  front  a  smooth  plane,  the  under  side  of  the  projecting  end 
of  the  roof  is  lighter  than  the  vertical  wall  because  it  is  so  situated 
as  to  receive  a  larger  proportion  of  reflected  light. 

7th.  Shadows  projected  from  circular  objects  are  also  generally 
well  defined,  but  the  shadings  instead  of  being  marked  by4>road 
bold  lines  as  they  are  in  rectangular  figures,  gradually  increase 
from  bright  light  to  the  darkest  shade  and  again  recede  as  the 
opposite  side  is  modified  by  the  reflections  from  surrounding  ob- 
jects, so  gradually  does  the  change  take  place  that  it  is  difficult 
to  define  the  exact  spot  where  the  shade  commences,  the  lights 
and  shades  appear  to  melt  into  each  other,  and  by  its  beautifully 
swelling  contour  enables  us  at  a  glance  to  define  the  shape  of  the 
object. 

8th.  DOUBLE  SHADOWS. — Objects  in  the  interior  of  buildings  fre- 
quently cast  two  or  more  shadows  in  opposite  directions,  as  they 
receive  the  light  from  opposite  sides  of  the  building;  this  effect  is 
also  often  produced  in  the  open  air  by  the  reflected  light  thrown 
from  some  bright  surface,  in  this  case  however,  the  shadow  from 
the  direct  rays  is  always  the  strongest;  in  a  room  at  night  lit  by 
artificial  means,  each  light  projects  a  separate  shadow,  the  strength 
of  each  depending  on  the  intensity  of  the  light  from  which  it  is 
cast,  and  its  distance  from  the  object;  the  student  may  derive 
much  information  from  observing  the  shading  and  shadows  of 
objects  from  artificial  light,  as  he  can  vary  the  angle,  object  and 
plane  of  projection  at  pleasure. 

9th.  The  extent  of  a  shadow  depends  on  the  angle  of  the  rays  of 
light.  If  we  have  a  given  object  and  plane  on  which  it  is  pro- 
jected, its  shadow  under  a  clear  sky  will  vary  every  hour  of  the 

— ^— — — '     "" '  ""        ' '        •       •  '••"• -•  !.!• •*.    —t^-J—J 


PLATE    LV.  113 

day,  the  sun's  rays  striking  objects  m  a  more  slanting  position  in 
the  morning  and  evening  than  at  noon,  projects  much  longer 
shadows.  But  in  mechanical  or  architectural  drawings  made  in 
elevation,  plan  or  section,  the  shadows  should  always  be  project- 
ed at  an  angle  of  45°,  that  is  to  say,  the  depth  of  the  shadow 
should  always  be  equal  to  the  breadth  of  the  projection  or  inden- 
tation; if  this  rule  is  strictly  followed,  it  will  enable  the  work- 
man to  apply  his  dividers  and  scale,  and  ascertain  his  projections 
correctly  from  a  single  drawing. 

NOTE. — The  best  method  for  drawing  lines  at  this  angle ;  is  to  use  with  the 
T  square,  a  right  angled  triangle  with  equal  sides,  the  hypothenuse  will  be  at 
an  angle  of  45°  with  the  sides ;  with  the  hypothenuse  placed  against  the  edge 
of  the  square,  lines  may  be  drawn  at  the  required  angle  on  either  side. 


PLATE    LV. 

PRACTICAL  EXAMPLES  FOR  THE  PROJECTION  OF  SHADOWS, 


FIGURE  1 

Is  a  square  shelf  supported  by  two  square  bearers  projecting  from 
a  wall.  The  surface  of  the  paper  to  represent  the  wall  in  all  the 
following  diagrams. 

1st.  Let  Jl.  B.  C.  D  be  the  plan  of  the  shelf;  Jl.  B  its  projection 
from  the  line  of  the  wall  W.  X;  B.  D  the  length  of  the  front  of 
the  shelf,  and  ,£and  F  the  plans  of  the  rectangular  bearers. 

2nd.  Let  G.  H  be  the  elevation  of  the  shelf  shewing  its  edge,  and 
J  and  K  the  ends  of  the  bearers. 

3rd.  From  all  the  projecting  corners  on  the  plan,  draw  lines  at  an 
angle  of  45°  to  intersect  the  line  of  the  wall  W.  X,  and  from 
these  intersections  erect  indefinite  perpendiculars. 

4th.  From  all  the  projecting  corners  on  the  elevation,  draw  lines  at 
an  angle  of  45°  to  intersect  the  perpendiculars  from  correspond- 
ing points  in  the  plan ;  the  points  and  lines  of  intersection  define 
the  outline  of  the  shadow  as  shewn  in  the  diagram. 

15  ~~ 


114  PLATE    LV. 

FIGURE  2 


Is  a  square  Shelf  against  a  wall  supported  by  two  square  Uprights. 

L.  M.  JV*.  0  is  the  plan  of  the  shelf,  P  and  Q  the  plans  of  the  up- 
rights, R.  S  the  front  edge  of  the  shelf,  T  and  V  the  fronts  of  the 
uprights. 

1st.  From  the  angles  on  the  plan  draw  lines  at  an  angle  of  45°  to 
intersect  W.  X,  and  from  the  intersections  erect  perpendiculars. 

2nd.  From  R  and  S,  draw  lines  at  an  angle  of  45°  to  intersect  the 
corresponding  lines  from  the  plan. 


FIGURE  3 


Is  a  Frame  with  a  semicircular  head,  nailed  against  a  wall,  the 
Frame  containing  a  sunk  Panel  of  the  same  form. 

1st.  Let  JL.  B.  C.  D  be  the  section  of  the  frame  and  panel  across 

the  middle,  and  F  on  the  elevation  of  the  panel,  the  centre  from 

which  the  head  of  the  panel  and  of  the  frame  is  described. 
2nd.  From  E,  draw  a  line  to  intersect  the  face  of  the  panel,  and 

from  D  to  intersect  W.  X,  and  erect  the  perpendiculars  as  shewn 

by  the  dotted  lines. 
3rd.  From  JY  and  JV*/  draw  lines  to  define  the  bottom  shadow, 

and  at  L  draw  a  line  at  the  same  angle  to  touch  the  curve. 
4th.  At  the  same  angle  draw  F.  G,  make  F.  H  equal  to  the  depth 

of  the  panel,  and  F.  G  equal  to  the  thickness  of  the  frame. 
5th.  From  H  with  the  radius  F.  R,  describe  the  shadow  on  the 

panel,  and  from  G  with  the  radius  F.  S,  describe  the  shadow  of 

the  frame. 
NOTE. — The  tangent  drawn   at  L  and  the  curve  of  the  shadow  touch  the 

edge  of  the  frame  in  the  same  spot,  but  if  the  proportions  were  different  they 

would  not  do  so ;  therefore  it  is  always  better  to  draw  the  tangent. 

FIGURE  4 


Is  a  Circular  Stud  representing  an  enlarged  view  of  one  of  the 
Nail  Heads  used  in  the  last  diagram,  of  which  N.  O.  P  is  a  sec- 
tion through  the  middle,  and  W.  X  the  face  of  the  frame. 

1st.  Draw  tangents  at  an  angle  of  45°  on  each  side  of  the  curve. 
2nd.  Through  L  the  centre,  draw  L.  JV/,  and  make  L.  M  equal  to 
the  thickness  of  the  stud. 


|]4  PLATE    LV. 

FIGURE  2 
Is  a  square  Shelf  against  a  wall  supported  by  two  square  Uprights. 

L.  M.  jY.  0  is  the  plan  of  the  shelf,  P  and  Q  the  plans  of  the  up- 
rights, R.  S  the  front  edge  of  the  shelf,  T  and  V  the  fronts  of  the 
uprights. 

1st.  From  the  angles  on  the  plan  draw  lines  at  an  angle  of  45°  to 
intersect  W.  X,  and  from  the  intersections  erect  perpendiculars. 

2nd.  From  R  and  S,  draw  lines  at  an  angle  of  45°  to  intersect  the 
corresponding  lines  from  the  plan. 

FIGURE  3 


Is  a  Frame  with  a  semicircular  heady  nailed  against  a  wall,  the 
Frame  containing  a  sunk  Panel  of  the  same  form. 

1st.  Let  Jl.  E.  C.  D  be  the  section  of  the  frame  and  panel  across 

the  middle,  and  F  on  the  elevation  of  the  panel,  the  centre  from 

which  the  head  of  the  panel  and  of  the  frame  is  described. 
2nd.  From  E,  draw  a  line  to  intersect  the  face  of  the  panel,  and 

from  D  to  intersect  W.  X,  and  erect  the  perpendiculars  as  shewn 

by  the  dotted  lines. 
3rd.  From  J\T  and  JV"/  draw  lines  to  define  the  bottom  shadow, 

and  at  L  draw  a  line  at  the  same  angle  to  touch  the  curve. 
4th.  At  the  same  angle  draw  F.  G,  make  F.  H  equal  to  the  depth 

of  the  panel,  and  F.  G  equal  to  the  thickness  of  the  frame. 
5th.  From  H  with  the  radius  F.  R,  describe  the  shadow  on  the 

panel,  and  from  G  with  the  radius  F.  S,  describe  the  shadow  of 

the  frame. 
NOTE. — The  tangent  drawn   at  L  and  the  curve  of  the  shadow  touch  the 

edge  of  the  frame  in  the  same  spot,  but  if  the  proportions  were  different  they 

would  not  do  so ;  therefore  it  is  always  better  to  draw  the  tangent. 

FIGURE  4 

Is  a  Circular  Stud  representing  an  enlarged  mew  of  one  of  the 
Nail  Heads  used  in  the  last  diagram,  of  which  N.  O.  P  is  a  sec- 
tion through  the  middle,  and  W.  X  the  face  of  the  frame. 

1st.  Draw  tangents  at  an  angle  of  45°  on  each  side  of  the  curve. 
2nd.  Through  L  the  centre,  draw  L.  M,  and  make  L.  M  equal  to 
the  thickness  of  the  stud. 


Plate  55. 
SHADOWS. 


Fig.  5. 


Mate,  56. 
SHADOWS. 


/'/,/.  7 


fflma.nkS.oi 


PLATE    LY. 


3rd.  From  Jkf,  with  the  same  radius  as  used  in  describing  the  stud, 
describe  the  circular  boundary  of  the  shadow  to  meet  the  two 
tangents,  which  completes  the  outline  of  the  shadow. 


FIGURE  5 

Is  a  Square  Pillar  standing  at  a  short  distance  in  front  of  the 

wall  W.  X. 

1st.  Let  A.  B.  C.  D  be  the  plan  of  the  pillar,  and  W.  X  the  front 
of  the  wall,  from  JL.  C.  D  draw  lines  to  W.  X,  and  from  their 
intersections  erect  perpendiculars. 

2nd.  Let  E.  F.  G.  #be  the  elevation  of  the  pillar,  from  F  draw 
F.  K.  L  to  intersect  the  perpendiculars  from  C  and  D. 

3rd.  Through  K,  draw  a  horizontal  line,  which  completes  the  out- 
line. The  dotted  lines  shew  the  position  of  the  shadow  on  the 
wall  behind  the  pillar. 


PLATE    LVI. 

SHADOW  S— C  0  N  T  I  N  U  E  D . 


FIGURE  1 

Is  the  Elevation  and  Fig.  2  the  Plan  of  a  Flight  of  Steps  with 
rectangular  Blockings  at  the  ends,  the  edge  of  the  top  step  even  with 
the  face  of  the  wall. 

1st.  From  A.  B.  Cand  D,  draw  lines  at  an  angle  of  45°. 

2nd.  From  F  where  the  ray  from  C  intersects  the  edge  of  the  front 
step,  draw  a  perpendicular  to  JV,  which  defines  the  shadow  on 
the  first  riser. 

3rd.  From  Q  where  the  ray  from  C  intersects  the  edge  of  the  se- 
cond step,  draw  a  perpendicular  to  M,  which  defines  the  shadow 
on  the  second  riser. 

4th.  From  K  where  the  ray  from  JL  intersects  the  top  of  the  third 
step,  draw  a  perpendicular  to  0,  which  defines  the  shadow  on  the 
top  of  that  step. 


116  PLATE    LVI. 

5th.  From  L  where  the  ray  from  A  intersects  the  top  of  the  second 
step,  draw  a  perpendicular  to  H  intersecting  the  ray  drawn  from 
C  in  Hy  which  defines  the  shadow  on  the  top  of  the  second  step. 

6th.  From  P  where  the  ray  from  B  intersects  the  ground  line, 
draw  a  perpendicular  tp  intersect  the  ray  drawn  from  D  in  E; 
this  defines  the  shape  of  the  shadow  on  the  ground. 


FIGURE     3. 
To  draw  the  Shadow  of  a  Cylinder  upon  a  Vertical  Plane. 

Rule. — Find  the  position  of  the  shadow  at  any  number  of  points. 

1st.  From  Jl  where  the  tangental  ray  (at  an  angle  of  45°)  touches 
the  plan,  draw  the  ray  to  W.  X,  and  from  the  intersection  erect  a 
perpendicular. 

2nd.  From  Jl  erect  a  perpendicular  to  B,  and  from  B  draw  a  ray 
at  45°  with  Jl.  B  to  intersect  the  perpendicular  from  A  in  L. 
This  defines  the  straight  part  of  the  shadow. 

3rd.  From  any  number  of  points  in  the  plan  E.  H,  draw  rays  to 
intersect  the  wall  line  W.  X,  and  from  these  points  of  intersec- 
tion erect  perpendiculars. 

4th.  From  the  same  points  in  the  plan  erect  perpendiculars  to  the 
top  of  the  cylinder,  and  from  the  ends  of  these  perpendiculars 
draw  rays  at  45°  to  meet  the  perpendiculars  on  the  wall  line;  the 
intersections  give  points  in  the  curve. 

NOTE  1. — The  outlines  of  shadows  should  be  marked  by  faint  lines,  and  the 
shadow  put  on  by  several  successive  coats  of  India  ink.  The  student  should 
practice  at  first  with  very  thin  color,  always  keep  the  camel  hair  pencil  full, 
and  never  allow  the  edges  to  dry  until  the  whole  shadow  is  covered.  The 
same  rule  will  apply  in  shading  circular  objects ;  first  wash  in  all  the  shaded 
parts  with  a  light  tint,  and  deepen  each  part  by  successive  layers,  always 
taking  care  to  cover  with  a  tint  all  the  parts  of  the  object  that  require  that 
tint;  by  this  means  you  will  avoid  harsh  outlines  and  transitions,  and  give 
your  drawing  a  soft  agreeable  appearance. 

NOTE  2. — The  lightest  part  of  a  circular  object  is  where  a  tangent  to  the 
curve  is  perpendicular  to  the  ray  as  at  P.  The  darkest  part  is  at  the  point 
where  the  ray  is  tangental  to  the  curve  as  at »/?,  because  the  surface  beyond 
that  point  receives  more  or  less  reflected  light  from  surrounding  objects. 


117 


THEORY  OF  COLOR, 


AND  ITS  APPLICATION  TO 


ARCHITECTURAL  AND   MECHANICAL 

DEALINGS. 


THE    THEORY. 

WHEN  we  survey  with  attention,  the  beautiful  coloring  of  the  works 
of  nature ;  we  cannot  fail  to  perceive  the  almost  infinite  variety 
of  tints  and  hues,  of  which  the  landscape  is  composed,  and 
however  its  tone  may  be  modified  by  the  state  of  the  atmosphere, 
by  the  changes  of  the  seasons,  or  by  the  degree  of  light  with 
which  it  is  illuminated,  we  shall  always  find  these  colors  blended 
or  contrasted  harmoniously;  forming  a  glorious  whole,  highly 
satisfactory  when  viewed  in  mass,  and  much  more  so  when 
analyzed  and  examined  in  detail. 

But  numerous  as  are  those  hues,  it  has  been  demonstrated  that  all 
are  composed  of  three  primary  colors,  viz. 

YELLOW,    RED    AND    BLUE. 

These  names,  however,  are  commonly  applied  to  various  tints  of 
the  colors,  and  therefore  do  not  convey  to  the  mind  a  sufficiently 
definite  idea ;  but  they  may  be  seen  in  their  pure  brilliancy,  in 
the  flowers  of  the  Yellow  Jasmine,  the  Red  Geranium  and  the 
Blue  Sage. 

We  seldom  find  them  used  in  their  intensity,  in  nature's  own  paint- 
ing ;  and  chiefly  on  the  smaller  gems  with  which  she  loves  to 
decorate  her  bosom,  from  which  the  above  examples  have  been 
selected,  and  even  here  they  are  used  so  sparingly  that  few  ex- 
amples of  the  pure,  unmixed,  primary  colors  can  be  found ;  so  es- 
pecially is  this  the  case  with  blue,  that  many  horticulturists  affirm 
that  a  perfectly  blue  flower  is  unknown  in  nature  ;  but  our  ex- 
ample will  give  a  tolerably  accurate  idea  of  the  color. 


118  THEOKY     OF      COLOR. 

On  a  cursory  view,  it  would  appear  almost  impossible  that  the 
blending  of  three  simple  elements  can  produce  so  great  variety  ; 
if  those  elements  could  only  be  used  as  wholes,  the  changes  would 
necessarily  be  very  limited  ;  but  let  us  endeavor  to  realize  the  fact, 
that  any  given  quantity  of  a  color  may  be  divided  and  subdivided, 
again  and  again,  into  very  minute  portions  ;  and',  that  any  minute 
quantity  may  be  blended  with  equal  or  larger  proportions  of  both 
or  either  of  the  other  colors,  and  the  proportions  of  each  or  all  of 
them  changed  at  pleasure,  and  that  each  combination  of  propor- 
tions will  modify,  to  some  extent,  the  hue  produced. 

To  illustrate  this  position,  let  us  suppose  that  we  have  three  fluids, 
yellow,  red  and  blue,  four  drops  of  each,  and  ascertain  by  calcula- 
tion how  many  different  combinations  of  distinct  hues  might  be 
produced  by  combining  them,  two  or  more  drops  together  to  con- 
stitute a  hue  ;  the  difference  between  each  combination  to  be  not 
less  than  a  drop. 

The  answer  will  be  eighty-eight,  and  if  we  add  the  three  original 
colors,  the  number  will  be  ninety-one. 

The  greatest  disparity  between  the  proportions  in  this  arrangement, 
would  be,  where  one  drop  of  a  color  is  mixed  with  four  drops  of 
each  of  the  others,  the  smallest  quantity  comprising  one-ninth  of 
the  whole  mixture;  enough  sensibly  to  modify  the  hue  produced. 

If  instead  of  four  drops  we  were  to  suppose  six  drops  of  each  given  ; 
by  the  same  process  we  could  produce  about  two  hundred  changes, 
and  the  greatest  disproportion  in  any  hue,  would  be  as  one  to  six 
of  each  of  the  others,  comprising  one-thirteenth  of  the  whole. 
After  this  very  limited  illustration,  let  us  consider  that  any  of 
these  distinct  hues  may  be  used  of  any  degree  of  depth,  from  the 
faintest  trace  of  color  to  its  deepest  intensity,  producing  innumera- 
ble tints,  and  we  shall  be  able  to  form  some  idea  of  the  truth,  that 
all  may  be  produced  by  three  original  colors. 

Before  proceeding  with  this  subject,  and  that  we  may  do  so  more 
understandingly,  it  will,  perhaps,  be  as  well  to  define  the  mean- 
ing of  some  of  the  terms  used,  and  first  of 

COLOR.  This  term  is  used  in  a  very  extended  sense.  It  may  be 
applied  to  any  tint  or  hue  produced  in  nature  by  all  its  varied 
processes;  and  with  equal  propriety,  to  all  the  pigments  and 
paints  used  by  Artists  and  Painters  to  imitate  them  ;  but  in  this 
branch  of  our  subject,  (the  theoretical,)  it  is  necessary  to  limit  its 
application  to  the  hues  produced  ;  therefore  when  we  speak  of 


THEORY     OF     COLOR.  119 

color,  we  do  not  mean  prussian  Hue,  ultra-marine,  or  any 
other  pigment  used  to  produce  blue  in  art,  but  simply  the  hue 
itself,  and  in  the  same  manner  of  all  the  other  colors. 

HUE.  This  term  is  synonymous  with  color,  when  it  is  used  as  re- 
stricted in  the  last  paragraph,  and  may  be  applied  to  any  tint  or 
color  produced  in  Nature  or  Art. 

Some  writers  apply  these  terms  more  restrictedly  ;  for  example, 
the  term  color  is  applied  to  the  primary  colors,  yellow,  red  and 
blue,  and  to  the  secondary  colors  produced  by  equal  admixture 
of  two  of  the  primaries,  viz :  orange,  green  and  purple,  and  the 
term  hue  is  applied  by  them  to  any  color  in  which  all  the  prima- 
ries are  mixed,  either  in  equal  or  unequal  quantities  ;  thus  a 
russet  hue,  an  olive  hue,  &c.;  but  in  all  cases,  where  any  definite 
color  is  meant  by  either  of  these  terms,  the  name  of  the  color  or 
hue  must  be  added  to  it,  to  convey  the  proper  idea  to  the  mind. 

TINT.  This  term  is  used  to  denote  the  depth  or  strength  of  a 
color,  as  a  dark  tint,  a  light  tint,  &c.  Any  primary  or  mixed 
color  may  be  used  in  its  deepest  intensity,  or  so  faintly  as  to  be 
hardly  distinguished  from  white,  or  with  any  degree  of  depth 
between  these  points ;  these  various  gradations  from  the  deepest 
color,  are  called  tints. 

SHADE,  This  term  is  applied  to  any  of  the  gradations  oi  a  color 
toward  black ;  as  a  tint  is  applied  as  above  defined,  to  any  c(  its 
gradations  toward  white. 

Shade  is  also  applied  to  all  the  gradations  from  pure  white  light  t<> 
bla<ik. 

TONE — is  applied  to  designate  the  prevailing  hue  of  a  landscape  or 
other  object  in  nature,  or  of  a  room,  picture,  or  other  work  of  art ; 
this  ruling  hue  is  called  its  tone  or  key.  Thus  a  tone  may  le 
warm  or  cold,  grave  or  gay,  lively  or  sombre,  as  the  different  hues 
and  lights  prevail, 

KEEPING.  A  picture  is  said  to  be  in  keeping,  when  all  the  details 
of  outline  and  shading  are  correct,  and  in  which  the  colors  are 
blended  or  contrasted  harmoniously  ;  and  is  said  to  be  out  of  keep- 
ing, when  these  requisites  have  not  been  attended  to. 

HARMONY.  A  picture  is  said  to  be  in  harmony,  when  all  the 
requisites  <of  tone  .and  keeping  have  been  observed ;  but  if  the 
contrasts  are  harsh* or  gaudy,  or  any  of  the  objects  introduced  in- 
congruous to  the  general  design,  it  is  said  to  be  inharmonious. 

Harmony  is  also  applied  to  the  local  arrangement  of  an  assemblage 


120  THEORY     OF     COLOE. 

of  colors,  which  may  be  harmonious  or  otherwise,  as  the  Mendings 
and  contrasts  are,  or  are  not,  in  consonance  with  natural  effects. 

MELODY  and  harmony  are  applied  to  colors  as  they  are  in  music. 
Melody  has  reference  to  the  consonance  or  harmonious  agreement 
of  notes  or  hues  following  each  other  in  continued  sequence  ;  and 
harmony  is  applied  to  the  perfect  agreement  of  all  the  parts 
of  a  composition  with  each  other,  both  in  Music  and  Painting. 
Without  melody  in  either  Art,  harmony  cannot  exist. 

Thus,  melody  or  melodizing  colors,  are  those  which  follow  a  certain 
natural  scale,  and  form  a  pleasing  sequence  of  hues,  as  they  do  in 
the  solar  spectrum. 

CONTRASTING  COLORS  are  those  which  by  their  opposite  qualities 
make  each  appear  more  brilliant  than  either  would  do,  if  unac- 
companied by  the  other ;  and  produce  a  more  pleasing  effect  on 
the  eye. 

Contrasting  colors  are  also  called  complementary  eolors;  because 
the  harmonious  contrast  to  any  color,  is  the  color  or  eolors  required 
to  be  added  to  it,  to  make  up  the  triad  constituting  white  light ; 
which  is  exemplified  in  the  following  optical  fact. 

If  the  eye  be  steadily  directed  for  some  time  to  any  small  colored 
object  placed  in  a  prominent  position,  it  will  appear  to  be  sur- 
rounded with  another  color  which  is  its  complementary  color  as 
defined  above. 

The  following  experiment  may  be  readily  made  ;  place  a  red  wafer 
in  the  middle  of  a  sheet  of  white  paper,  and  look  at  it  steadily,  it 
will  soon  appear  to  be  surrounded  with  a  ring  of  pale  green,  and  if 
the  wafer  be  removed,  or  the  eye  directed  to  another  part  of  the 
paper,  a  circular  green  spot  will  appear ;  which  will  fade  away  as 
the  impression  made  by  the  red  wafer  passes  from  the  optic  nerve. 
If  a  green  wafer  be  substituted,  the  complementary  color  will  be 
red,  if  a  blue  wafer,  it  would  be  surrounded  with  orange. 

The  complementary  or  contrasting  color  of  a  primary  is  always  a 
secondary  color  compounded  of  the  other  two  primaries.  The 
complementary  of  a  secondary  color,  is  the  primary  which  does 
not  enter  into  its  composition. 

The  contrasting  color  of  a  tertiary  or  broken  color ;  is  a  similarly 
constituted  hue,  in  which  the  opposite  primaries  preponderate  in 
the  same  proportions,  making  up  the  harmonious  triad  as  above 
described. 

LIGHT,  SHADE  and  SHADOW,  have  already  been  treated  of  at  pages 


THEOEY     OF     COLOE.  121 

81,  82,  111  and  112,  of  the  Drawing  Book.  I  would  advise  the 
student  to  read  again  the  explanations  there  given,  to  aid  him  in 
understanding  some  of  the  experiments  that  have  been  made  with 
rays  of  light,  from  which  our  present  theory  of  colors  has  been 
derived. 

Paragraph  5,  page  82,  says :  When  a  ray  of  light  passes  from  a 
rare  to  a  more  dense  medium,  it  is  bent  out  of  its  direct  course, 
dc.,  this  effect  is  called  refraction.  Experiments  have  proved, 
that  a  ray  of  white  light  from  the  sun,  is  not  homogeneous  in  its 
composition  ;  that  some  portions  of  it  are  capable  of  being  re- 
fracted or  bent  out  of  a  direct  course,  to  a  much  greater  angle  than 
other  portions  of  the  same  ray,  and  when  so  refracted,  the  ray  is 
dissected  and  exhibits  several  different  colors  ;  this  discovery  we 
owe  to  Sir  Isaac  Newton,  who  made  the  following  experiment : 
In  a  darkened  room,  he  caused,  a  small  hole  to  be  bored  in  a  window- 
shutter,  through  which,  a  ray  of  light  from  the  sun  could  pass 
freely  in  a  straight  line,  which  falling  on  a  white  screen  placed  for 
the  purpose,  exhibited  a  circular  spot  of  white  light  corresponding 
in  diameter  to  the  hole  in  the  shutter,  appearing  the  more  brilliant, 
from  its  contrast  with  the  surrounding  darkness. 
He  then  interposed  a  triangular  prism  made  of  colorless  glass,  which 
receiving  the  ray  on  one  of  its  sides,  refracted  it  from  its  direct 
course  toward  the  upper  portion  of  the  screen,. forming  an  elongated 
image  of  the  sun,  composed  of  seven  different  colors  of  the  most 
intensely  brilliant  character,  arranged  in  the  following  order,  com- 
mencing at  the  top : 

VIOLET, 

INDIGO, 

BLUE, 

GREEK, 

YELLOW, 

ORANGE, 

RED. 

This  elongated  image  he  called  the  "SoLAE  SPECTRUM  ;"  the  elonga- 
tion is  caused  by  the  different  degree  of  refrangibility  of  the  colors 
of  which  the  ray  was  composed.  Thus  red  is  less  refrangible  than 
orange — orange  than  yellow,  and  so  on  to  violet,  which  is  the  most 
refrangible. 


16 


122  THEORY     OF     COLOE. 

In  continuance  of  this  beautiful  experiment,  Sir  Isaac,  by  a  series 
of  reflectors  placed  in  the  spectrum,  threw  all  the  colors  to  the 
same  point,  where  they  were  blended  together  again,  and  the  ray 
restored  to  its  pure  white  character. 

From  this  and  other  experiments  he  arrived  at  the  conclusion,  that 
light  was  composed  of  seven  distinct  homogeneous  colors. 

It  was  soon  noticed,  however,  by  those  who  repeated  the  above  ex- 
periment, that  the  colors  had  no  positive  line  of  separation,  but 
that  the  adjoining  colors  were  gradually  blended  together :  thus 
the  red  changed  by  almost  imperceptible  gradations  to  orange ;  the 
orange  to  yellow,  the  yellow  to  green,  and  the  green  to  blue,  and 
so  on. 

This  led  them  to  suppose  that  as  orange  appeared  between  the  red 
and  the  yellow,  it  was  not  homogeneous,  but  was  formed  by  the 
blending  of  the  adjoining  colors  in  equal  quantities ;  that  green 
was  formed  in  like  manner  by  the  blending  of  the  yellow  and  the 
blue,  that  the  various  hues  of  purple  and  violet,  may  be  formed  by 
the  blending  of  red  with  blue,  and  that  consequently  there  are  but 
three  primary  colors,  yellow,  red  and  blue. 

This  theory,  agreed  with  the  theory  held  by  painters,  who  affirmed 
that  all  the  hues  and  tints  of  nature  can  be  successfully  imitated 
with  those  three  colors,  not  excepting  the  beauteous  rainbow,  which 
exhibits  all  the  colors  of  the  solar  spectrum,  arranged  in  the  same 
relative  positions,  and  formed  in  like  manner  by  the  refraction  of 
light  through  the  drops  of  the  falling  shower :  they  also  asserted, 
that  yellow,  red  and  blue,  cannot  be  formed  by  the  mixture  of  any 
other  colors ;  and  that,  as  all  other  colors  can  be  formed  by  the 
mixture  of  these  three,  they  are,  therefore,  the  only  homogeneous 
colors. 

Later  experiments  have  proved  this  theory  to  be  correct. 

The  most  elaborate  of  the  later  experiments  were  made  by  Sir  David 
Brewster ;  he  demonstrated,  by  experimenting  with  rays  of  the 
different  colors  passed  through  a  prism,  in  the  same  manner  as  in 
Newton's  experiment,  that  each  color  is  projected  on  the  screen,  of 
the  same  length  as  the  spectrum  of  the  white  ray. 

Thus,  if  a  red  ray  be  passed  through  the  prism,  an  oblong  red  image 
will  be  projected  on  the  screen,  occupying  under  the  same  circum- 
stances, the  same  space  as  the  entire  spectrum  ;  but  not  all  equally 
red,  as  the  color  is  not  equally  distributed,  the  greatest  accumu- 


THEORY     OF     COLOE.  123 

lation  of  color  being  near  the  bottom,  at  a  point  that  would  be  the 
centre  of  the  red  in  the  spectrum. 

A  ray  of  yellow  forms  a  yellow  image  of  the  same  size,  also  un- 
equally distributed,  the  most  intense  yellow  being  at  the  centre  of 
the  yellow  in  the  spectrum,  gradually  becoming  fainter  toward  the 
ends. 

A  blue  ray  forms  in  like  manner  a  blue  image  of  the  same  length, 
the  greatest  accumulation  of  color  being  still  higher  on  the  spectrum, 
at  the  point  where  the  blue  and  indigo  meet. 

These  experiments  prove  conclusively,  that  the  spectrum  is  formed 
of  three  homogeneous  primary  colors,  and  that  the  violet,  indigo, 
green  and  orange,  result  from  the  unequal  blending  of  the  three 
primaries. 

In  addition  to  the  three  primary  colors,  we  must  add"  to  the  scale, 
LIGHT  and  its  absence,  SHADE  ;  which  are  represented  by  white 
and  black. 

Black,  in  the  strictest  application  of  the  term,  is  the  total  absence  of 
light,  and  must  be  invisible ;  and  as  the  blackest  object  that  we 
can  perceive  must  reflect  some  faint  ray  of  light  to  make  it  visible, 
it  must  have  somewhat  of  a  grayish  tint ;  gray  being  the  mean 
between  light  and  shade.  The  term  black  is  usually  applied  to 
the  darkest  shades  either  in  Nature  or  Art. 

We  have  learnt  from  the  foregoing  experiments,  that  white  light 
may  be  resolved  into  the  three  primary  colors  ;  these  three  primaries 
mixed  in  definite  proportions  will  produce  black  ;  therefore,  neither 
white  nor  black  can  be  considered  as  a  primary  homogeneous  color  ; 
and  in  fact,  are  not  theoretically  considered  as  colors. 

The  term  color,  however,  is  applied  to  the  white  and  black  pigments 
or  paints,  used  to  produce  light  and  shade  in  works  of  art. 

The  following  may  be  called  the  primary  scale  retreating  from  light 
to  shade. 

WHITE, 

YELLOW, 

EED, 

BLUE, 

BLACK. 

A  union  of  any  two  of  the  primaries  in  equal  quantities  gives  a 
secondary  color. 


124 


THEOEY     OF     COLOE. 


The  secondaries  are  also  three  in  number,  as  follows : 

Yellow  and  Red  produce  the  secondary  OEANGE. 
Yellow  and  Blue          "  "         GEEEN. 

Eed  and  Blue  "  "         PUEPLE. 

In  like  manner,  three  tertiary  colors  are  produced  by  the  union  of 
the  secondaries ;  consequently,  in  the  composition  of  the  tertiaries, 
all  the  primaries  enter  in  unequal  proportions. 

They  are  compounded  as  follows  : 


produce   OITEON 

or 
YELLOW  HUE. 

produce  RUSSET 

or 
RED  HUE. 

produce  OLIVE 

or 
BLUE  HUE. 


2  parts  Yellow, 
1       "      Red. 
1      "      Blue. 

1  part  Yellow. 

2  '•    Red. 
1      "    Blue. 

1  part  Yellow. 

1  "    Red. 

2  "    Blue. 


It  will  thus  be  perceived  that  a  tertiary  color  is  composed  of  two 
parts*  or  equivalents  of  one  of  the  primaries,  with  one  part  of  each 
of  the  others ;  and  each  occupies  its  place  in  the  following  scale,  as 
a  light  or  a  dark  color  predominates : 

WHITE. 
Primary.         YELLOW. 

OEANGE.         Secondary. 
Primary.         RED. 

PUEPLE.          Secondary. 
Primary.         BLUE. 

GEEEN.  Secondary. 

CITEON,  I 

RUSSET,   [       Tertiaries. 

OLIVE,    J 

BLACK. 

*In  speaking  of  parts  in  this  connection  we  must  understand,  equal  quantities  as  to  power.  Field 
in  his  Chromatics  says,  "three  parts  yellow,  will  neutralize  five  parts  red  or  eight  parts  blue, 
and  if  a  circular  disk  be  colored  in  ftiese  proportions  from  the  centre  to  the  circumference,  and 
be  made  to  revolve  rapidly,  the  different  colors  will  be  blended  and  the  disk  will  appear  of  a 
dull  white." 


THEORY     OF     COLOK.  125 

In  this  scale  the  primaries  are  arranged  as  before,  in  regular  grada- 
tion from  light  to  shade ;  the  secondary  colors  are  placed  between 
the  primaries  from  which  they  are  derived,  and  these  are  followed 
in  due  order  by  the  tertiaries,  forming  a  natural  scale  of  easy 
reference,  which  we  shall  follow  in  describing  the  individual  colors. 

The  following  arrangement  exhibits  all  the  colors  in  continued 
sequence  from  light  to  shade. 


Advancing  colors. 


WHITE. 

YELLOW, 

ORANGE, 

CITRON, 

GREEN, 

RED,  The  middle  or  neutral  point. 

RUSSET, 

'       \  Retreating  colors. 
BLUE, 

PURPLE, 
BLACK. 

Advancing  colors  are  those  which  advance  from  the  neutral  red 
toward  light ;  and  when  applied  to  objects  in  a  picture  or  other 
composition,  they  cause  them  to  appear  near  the  spectator. 

Retreating  colors  are  those  which  retreat  from  the  neutral  point 
toward  shade,  and  when  applied  to  objects  will  cause  them  to  re- 
cede or  retreat  from  the  spectator.  If  we  should  color  two  similar 
objects  placed  beside  each  other  in  a  composition,  the  one  yellow 
and  the  other  purple^  of  the  same  strength  of  tint,  the  yellow  object 
will  appear  to  be  in  advance  of  the  purple ;  and  the  same  effect  is 
produced,  but  in  a  less  degree,  by  the  other  advancing  and  retreat- 
ing colors,  yellow  and  purple  being  at  the  extremes  of  the  scale. 

Although  pure  RED  is  the  neutral  point  between  light  and  shade, 
it  is  in  one  respect  a  decidedly  advancing  color ;  with  the  same 
strength  of  tint  it  causes  objects  to  appear  in  advance  of  every 
other  color  except  orange  and  yellow ;  in  this  respect  green  and 
russet  should  also  change  places  in  the  scale. 

Colors  that  advance  objects  are  of  a  warm  tone  in  which  yellow  and 
red  prevail ;  and  those  which  cause  them  to  recede  are  of  a  cool 
tone  in  which  blue  is  always  found.  Very  distant  objects  generally 
a]  pear  of  a  bluish  gray  ;  in  mountainous  countries  we  may  gener- 
ally find  a  blue  ridge,  the  name  being  derived  from  this  cause. 


126  THEOEY     OF     COLOK. 

This  advancing  or  retreating  effect  is  much  aided  by  graduating  the 
strength  of  the  tints  in  a  picture,  distant  objects  requiring  lighter 
tints  than  those  which  are  near. 

WHITE, 

representing  the  pure  light  of  the  sun,  is  placed  at  the  top  of  the 
scale.  Light  is  always  productive  of  more  cheerful  feelings  than 
darkness ;  the  same  effect  is  also  produced  by  its  representative. 
White  contrasts  harmoniously  with  all  the  colors,  but  its  most 
perfect  contrast  is  black :  it  also  melodizes  with  all  the  colors ;  but 
yellow,  which  is  nearest  allied  to  light,  is  its  melodizing  color. 
When  pure  white  is  placed  beside  a  deep  tint  of  any  of  the  colors 
except  yellow,  it  is  more  or  less  affected  by  it,  appearing  like  a 
light  tint  of  the  color  with  which  it  is  in  contact :  this  effect  is  not 
produced  by  yellow. 

EFFECTS  OF  REFLECTIONS  FROM  COLORED  SURFACES-  AND  OF 

ARTIFICIAL  LIGHT. 

Undecomposed  light  from  the  heavenly  bodies  being  pure  white, 
doe-s  not  alter  the  hues  of  any  of  the  colors  when  it  is  received 
direct ;  but  when  it  is  reflected  from  any  strongly  colored  surface, 
it  does  not  retain  its  purity,  but  partakes  of  the  color  of  the  reflect- 
ing surface ;  a  reflected  ray  of  light  therefore,  except  from  a  white 
surface,  is  a  ray  of  colored  light,  and  will  to  some  extent  modify 
the  hue  of  any  object  on  which  it  is  projected ;  thus  if  a  ray  from 
a  bright  red  surface  be  reflected  on  a  blue  object,  its  hue  will  be 
modified  toward  purple  ;  or,  if  a  ray  from  a  blue  surface  be  thrown 
on  yellow,  the  hue  will  incline  toward  green  :  of  course  the  amount 
of  such  modification  will  depend  upon  the  intensity  and  quality  of 
the  tints  of  both  the  reflecting  and  receiving  surfaces. 

The  light  produced  for  artificial  illumination  by  the  burning  of  gas, 
oil,  <£c.,  always  inclines  to  yellow,  and  has  the  same  effect  in 
modifying  colors  as  the  reflected  ray  from  a  colored  surface  :  hence 
it  is  very  difficult  to  distinguish  blue  from  green  by  artificial  light, 
as  the  yellow  rays  falling  on  the  blue,  will  naturally  produce  the 
secondary  green ;  the  other  colors  are  also  modified  by  ai  tificial 
light,  but  not  so  remarkably. 

This  effect  of  artificial  light  should  always  be  taken  into  considera- 
tion in  the  execution  of  paintings  and  in  the  decoration  of  rooms 
that  are  intended  to  be  chiefly  viewed  or  used  at  night :  such  as 


THEORY     OF     COLOR.  127 

Panoramas,  or  the  scenery  and  decorations  of  Theatres,  Ball  Rooms, 
&c.  Foliage,  for  example,  intended  to  be  viewed  under  these  cir- 
cumstances, requires  to  have  more  blue  in  its  composition  than  if  it 
were  to  be  viewed  by  daylight ;  for  a  green  that  would  represent 
the  vigorous  foliage  of  summer  by  daylight,  would  appear  to  have 
put  on  some  of  its  autumnal  characteristics  at  night,  and  a  green 
that  would  by  day  represent  the  hues  of  autumn,  would  by  artifi- 
cial light  degenerate  into  the  sere  and  yellow  leaf. 

ABSORPTION  AND  REFLECTION  OF  LIGHT,  TRANSPARENCY,  OPACITY. 

Paragraph  6,  page  82,  says :  "A  portion  of  light  is  absorbed  by  all 
bodies  receiving  it  on  their  surface,"  (fee.  See  paragraphs  6,  7  and 
8  as  above. 

This  fact  has  led  to  the  theory  that  bodies  Jiave  no  inherent  color ; 
that  different  substances  absorb  differently  colored  rays  of  light, 
which  are  transmitted  through  their  substance  ;  and  that,  conse- 
quently, the  unabsorbed  portion  of  the  ray  which  is  reflected, 
determines  the  color  of  the  surface.  Thus  if  we  suppose  the  entire 
red  of  a  ray  to  be  absorbed,  the  yellow  and  blue  would  be  reflected 
and  the  surface  would  appear  green ;  and  if  a  sufficiently  thin 
lamina  be  cut  from  this  green  substance  it  will  transmit  red  light. 
Therefore  the  color  of  any  substance,  depends  on  its  capacity  for 
reflecting  or  absorbing  the  differently  colored  rays ;  and  in  all 
cases,  the  color  of  the  light  transmitted  through  bodies,  is  composed 
of  the  complementary  colors  to  those  which  are  reflected,  and  give 
hue  to  its  surface. 

It  has  been  proved  by  experiment  that  the  most  dense  substances 
will  transmit  both  light  and  color  when  reduced  to  sufficiently 
thin  laminse  :  consequently  there  is  no  such  thing  known  in  nature 
as  a  really  opaque  substance. 

A  perfectly  opaque  substance  would  reflect  all  the  light  which  falls 
on  its  surface,  and  one  that  would  be  perfectly  transparent  would 
transmit  all  that  it  receives ;  but  these  perfect  conditions  are  not 
to  be  found. 

Transparency  and  Opacity  are,  then,  not  absolute  but  relative 
terms. 

Transparent  Media,  such  as  air,  water,  glass,  (fee.,  are  those  which 
permit  light  and  vision  to  pass  freely  through  them.  Translucent 
substances  are  those  which  are  not  sufficiently  transparent  to  allow 


128  THEORY     OF     COLOK. 

us  to  see  objects  through  them,  but  which  transmit  light  more  or 
less  freely ;  we  have  learnt  above,  that  the  most  dense  substances 
when  sufficiently  thin,  will  transmit  light ;  this  term  is  therefore 
very  indefinite,  a  substance  being  translucent  or  otherwise  as  its 
thickness  may  be  greater  or  less ;  a  leaf  of  this  book  if  placed 
before  an  opening  for  the  admission  of  light,  would  permit  so  much 
to  pass  through,  as  to  allow  objects  to  be  distinctly  visible,  and 
would  be  considered  highly  translucent,  but  close  the  book  com- 
posed of  numerous  leaves  of  the  same  material  and  place  it  in  the 
same  position,  it  would  exclude  the  light  and  be  properly  termed, 
opaque. 

YELLOW 

is  the  brightest  of  the  primary  colors  ;  it  is  contrasted  by  purple, 
which  is  composed  of  the  other  two  primaries.  With  red  it  forms 
the  secondary  orange,  and  with  blue  the  secondary  green ;  these 
are  its  melodizing  colors.  It  is  a  warm  color,  and  by  its  bright- 
ness it  heightens  the  effect  of  a  warm  arrangement.  Field,  in  his 
Grammar  of  Coloring,  says,  "The  sensible  effects  of  yellow  are 
gay,  gaudy,  glorious,  full  of  lustre,  enlivening  and  irritating ;  and 
its  impressions  upon  the  mind  partake  of  these  characters,  and 
acknowledge  also  its  discordances." 

But  these  effects  may  be  much  modified  in  art,  by  the  texture  of 
the  fabric  to  which  it  is  communicated ;  on  fine  and  rich  textures 
its  gay  effects  as  above  described  are  universally  recognized ;  we 
rarely  read  a  description  of  any  gorgeous  scene  without  finding  the 
purple  and  gold  named  as  the  reigning  colors ;  and  this  popular 
coupling  of  the.  theoretically  contrasting  colors,  may  be  taken  in 
evidence  of  the  correctness  of  the  theory.  The  character  and  effects 
of  this  color  are  entirely  changed  when  it  is  applied  to  coarse  fabrics. 
Goethe  says  :  *"When  a  yellow  color  is  communicated  to  dull  and 
coarse  surfaces,  such  as  common  cloth,  felt,  or  the  like,  on  which  it 
does  not  appear  with  full  energy,  the  disagreeable  effect  is  ap- 
parent. 

"By  a  slight  and  scarcely  perceptible  change  the  beautiful  impres- 
sion of  fire  and  gold  .is  transformed  into  one  not  undeserving  the 
epithet  foul,  and  the  color  of  harmony  and  joy  reversed  to  that  of 
ignominy  and  aversion." 

Yellow  should  never  be  used  in  large  masses  either  in  art  or  dec- 
Goethe's  Treatise  on  Colors,  extracted  from  Hay  on  the  Laws  of  Harmonious  Coloring, 


THEORY     OF     COLOR.  129 

oration.  When  it  is  judiciously  used  to  heighten  and  brighten  the 
effect  of  other  colors  it  has  a  gay  appearance,  -but  when  used  in 
large  masses  its  effect  is  always  gaudy. 

ORANGE, 

The  first  of  the  secondary  colors  in  the  scale,  is  compounded  of 
equivalents  of  yellow  and  red,  and  these  are  its  melodizing  colors  : 
its  contrasting  color  is  blue,  the  primary  which  does  not  enter  into 
its  composition. 

Orange,  being  compounded  of  red  the  color  of  fire,  and  yellow  the 
color  of  flame,  is  at  the  highest  point  of  warmth  in  coloring.  The 
contrast  between  orange  and  blue  is  more  powerful  than  that  be- 
tween any  other  two  colors. 

Orange,  like  yellow,  should  be  used  sparingly  and  for  the  same 
reasons ;  its  general  effect  in  a  composition  is  to  give  to  it  richness 
and  mellowness,  but  in  large  masses  it  is  gaudy  and  offensive  to 
the  eye. 

BED, 

In  its  relation  to  light  and  shade,  occupies  a  middle  place  between 
yellow  and  blue ;  its  melodizing  colors  are  orange  and  purple  which 
are  formed  by  its  union  with  the  other  primaries ;  its  contrasting 
color  is  green.  Red  rays  of  light  are  the  least  refrangible  of  the 
series,  and  occupy  the  lower  portion  of  the  solar  spectrum. " 

Red  is  a  warm,  powerful  color,  and  communicates  warmth  to  all 
works  in  which  it  predominates;  it  is  rendered  more  intensely 
warm  by  its  union  with  yellow,  but  its  tone  is  cooled  in  connection 
with  blue. 

Red  when  combined  with  yellow  or  blue  produces  a  series  of  bril- 
liant hues  before  arriving  at  the  secondaries  orange  and  purple : 
the  most  brilliant  of  these  is  SCARLET  in  its  combination  with 
yellow,  and  CRIMSON  when  mellowed  with  blue;  all  the  hues  be- 
tween these  are  popularly  known  as  red. 

Red  occupies  the  middle  or  neutral  point  between  light  and  shade; 
all  its  hues  in  combination  with  yellow,  through  scarlet  and  orange, 
and  through  yellow  up  to  white,  are  called  advancing  colors.  And 
all  the  hues  from  red  in  its  combinations  with  blue,  through  crim- 
son and  purple  toward  shade,  are  relatively  retreating  colors.  See 
scale  at  page  125. 

17 


130  THEORY     OF     COLOR. 

Red  is  the  most  positive  of  the  colors,  and  enters  largely  into  the 
composition  of  a  majority  of  the  hues  of  nature,  so  much  so  that 
neither  sky,  water  nor  foliage  can  be  successfully  imitated  without 
its  use;  but  in  nature,  it  is  rarely  seen  in  its  pure  unmixed  state  in 
its  deepest  tint ;  and  never  in  large  masses. 

In  the  Floral  world  where  it  is  chiefly  found,  it  is  invariably  accom- 
panied with  its  complementary  color,  green;  which  serves  to 
heighten  its  effect,  and  at  the  same  time  relieves  the  eye  from  its 
glare  and  adds  much  to  its  beauty. 

PURPLE 

Is  the  darkest  of  the  secondary  colors;  when  compounded  of  red 
and  blue  in  proportion  of  five  to  eight,  it  forms  the  contrasting 
color  to  pure  yellow;  red  and  blue  being  its  melodizing  colors ; 
it  has  a  pleasing  effect  on  the  eye,  is  the  most  retiring  of  colors, 
and  most  nearly  allied  to  shade.  A  compound  color  may  have 
numerous  hues,  as  either  of  the  primaries  predominate  in  different 
degrees  in  its  composition,  and  although  they  bear  the  same  gen- 
eral name,  yet  many  of  them  have  popular  specific  names,  thus 
purple  by  further  admixture  with  red  passes  through  various  hues 
into  crimson;  and  by  adding  blue  it  passes  into  violet. 
As  purple  is  a  pleasing,  and  unobtrusive  color,  it  may  be  used  in 
decoration  in  large  masses ;  but  its  use  should  be  confined  to  dec- 
orations intended  to  be  viewed  by  daylight,  as  the  yellow  of  arti- 
ficial light  neutralizes  and  destroys  its  effect. 

BLUE 

Is  the  representative  of  coldness  in  coloring  as  its  complementary 
color,  orange,  is  the  representative  of  heat;  with  yellow  it  forms 
the  secondary  green  and  with  red  the  secondary  purple,  these  are 
its  melodizing  colors,  but  its  accordance  with  those  colors,  more 
especially  with  green,  is  much  less  perfect  than  that  of  either  of 
the  other  primaries  with  its  secondaries.  Hay,  in  his  Laws  of 
Harmonious  Coloring,  says,  that  the  discordance  between  blue  and 
green  is  so  great  that  it  requires  a  neutral  gray  to  be  placed  be- 
tween them  to  produce  the  requisite  harmony  of  effect.  Blue  is 
the  most  retiring  of  the  primary  colors  and  most  allied  to  shade  ; 
it  is  a  brilliant,  beautiful  and  soothing  color,  but  it  requires  a  bright 
light  to  display  its  full  brilliancy,  in  a  dim  light  it  inclines  to  gray, 
and  artificial  light  makes  it  appear  green;  in  its  lighter  tints,  as  in 


THEORY     OF     COLOR.  131 

the  sky,  it  occupies  a  large  space  in  the  coloring  of  nature ;  when 
used  in  its  intensity  in  connection  with  the  other  primary  colors, 
it  requires  a  surface  equal  to  that  of  both  the  others  to  neutralize 
them,  as  in  the  experiment  of  the  circular  disk  for  producing  white, 
referred  to  at  page  124.  We  may  safely  follow  nature  in  this 
respect  and  employ  blue  with  good  effect  on  large  surfaces,  but  its 
predominance  for  decoration  or  for  a  picture,  either  in  intensity  or 
surface,  in  a  pure  or  mixed  state,  as  in  olive,  violet,  &c.,  will  always 
produce  a  cool  tone.  Next  to  black,  blue  forms  the  most  perfect 
contrast  to  white. 

GREEN, 

Although  the  last  of  the  secondaries  on  the  scale,  occupies,  in  its 
relation  to  light  and  shade,  a  middle  place  between  orange  and 
purple;  it  is  composed  of  yellow  and  blue  in  proportions  of  three 
to  eight,  these  are  its  melodizing  colors;  its  complementary  color  is 
red;  unlike  the  other  secondaries,  all  the  hues  compounded  of 
yellow  and  blue  bear  the  same  specific  name, — all  are  green; 
if  yellow  predominates,  it  is  called  a  yellow  green,  and  if  blue,  it 
is  called  a  blue  or  bluish  green. 

Green  is  the  almost  universal  color  of  vegetation — it  is  more  gener- 
ally diffused,  and  is  more  pleasing  and  agreeable  to  the  eye  than 
any  other  positive  color;  like  blue,  it  requires  a  bright  sunlight  to 
develop  its  full  brilliancy;  its  effect,  under  the  clear  sky  of  a  sum- 
mer day,  is  cool  and  refreshing,  as  it  reflects  but  little  light  and 
thus  relieves  the  eye  from  the  intense  rays  of  the  sun.  • 

When  yellow  predominates  in  its  composition,  it  is  slightly  an 
advancing  color;  when  it  inclines  toward  blue,  its  tone  is  cool  and 
retiring.  Its  effect  in  decoration  is  cheerful  and  gay,  but  when 
relieved  by  gold  and  contrasted  by  other  rich  warm  colors,  it  may 
be  called  brilliant;  for  artificial  light,  it  should  be  well  toned 
toward  blue. 

CITRON,  CITRINE  OR  YELLOW  HUE 

Is  the  brightest  of  the  tertiary  colors,  and  most  nearly  allied  to 
light;  it  is  compounded  of  orange  and  green  in  equal  proportions 
as  to  power,  consequently,  yellow  predominates  in  its  composition ; 
its  contrasting  color  is  dark  purple;  in  its  relations  to  other  colors, 
cjtron  may  be  considered  as  a  subdued  yellow. 


132  THEOKY     OF     COLOK. 

EUSSET  OR  RED  HUB 

Is  compounded  of  orange  and  purple;  and,  as  its  name  implies, 
its  ruling  tone  is  red,  which  enters  equally  into  the  composition  of 
each  of  the  secondaries  of  which  it  is  composed. 
Eusset  is  a  warm  and  pleasing  color,  occupying  of  the  tertiaries  the 
middle  space  between  light  and  shade;  its  contrasting  color  is  a 
deep  green.  The  different  hues  of  russet  are  commonly  called 

brown. 

OLIVE  OR  BLUE  HUE 

Compounded  of  green  and  purple,  is  the  last  and  darkest  of  the 
tertiaries  and  most  nearly  allied  to  shade;  its  complementary 
color  is  a  deep  orange?  blue  predominates  in  its  composition  in 
proportion  of  two  equivalents  of  blue  to  one  of  each,  of  the  other 
primaries;  it  is  a  cool  and  retiring  color,  soft  and  pleasing  in  its 
effects. 

The  tertiary  and  broken  colors  may  be  found  in  much  broader 
masses  in  both  Nature  and  Art,  than  either  the  primary  or  secon- 
dary colors;  being  more  subdued  in  their  character,  they  are  more 
soft  and  pleasing,  arid  serve  by  their  varied  contrasts  to  heighten 
the  effect  and  to  harmonize  the  more  brilliant  hues. 

BLACK, 

The  representative  of  darkness  and  contrast  of  white,  is  at  the 
bottom  of  the  scale ;  all  its  gradations  from  white  are  called  shades. 
Pure  black  when  compounded  of  the  three  primary  colors  in  their 
neutralizing  proportions  of  3,  5  and  8,  shows  no  trace  of  either  of 
the  colors  of  which  it  is  composed,  it  is  then  called  NEUTRAL 
BLACK,  and  all  its  gradations  up  to  white  are  called  NEUTRAL 
GRAYS. 

BROKEN  COLORS. 

This  term  is  applied  by  some  writers  to  all  colors  except  the 
primaries;  but,  in  its  general  acceptation,  it  is  applied  to  all  com- 
binations of  the  three  primaries  in  any  other  than  the  definite 
proportions  which  form  the  tertiaries,  and  also  in  Art  to  all  com- 
binations of  the  colors  with  black;  as  the  proportions  and  combi- 
nations of  the  primary  elements  may  be  almost  infinite  in  their 
variety  and  gradations,  it  would  be  impossible  to  give  distinctive 
names  to  the  various  hues. 


THEORY     OF     COLOR.  133 

These  colors  come  under  one  general  appellation,  that  of  SEMI- 
NEUTRALS;  occupying  a  position  between  the  positive  colors  of  the 
scale  and  the  neutral  black  or  gray.  They  are  al-so  further  divided 
into  three  classes,  each  class  governed  by  the  predominance  of  one 
of  the  primary  colors  in  its  composition :  they  are  BROWNS, 
MAROONS  and  GRAYS,  the  first  and  last  being  contrasts  to  each 
other  with  equal  and  opposite  strength  of  tints. 

The  first  term,  brown,  is  commonly  applied  to  all  broken  colors  of 
a  warm  tone  in  which  yellow  or  red  prevail,  but  it  should  be  re- 
stricted to  those  in  which  yellow  is  the  ruling  hue;  brown  is 
therefore  more  nearly  allied  to  light  than  either  of  the  other  classes, 
maroon  in  which  red  prevails  occupies  a  middle  place,  and  the 
semi-neutral  gray  is  most  nearly  allied  to  shade. 

Each  of  these  classes  includes  a  great  variety  of  hues,  many  of  them 
possessing  proper  names  peculiar  to  themselves,  thus  among  browns 
we  have  auburn,  hazel,  dun,  &c.;  in  the  second  class,  we  have 
puce,  murrey,  <&c.,  and  among  the  grays  we  have  slate,  ash,  lead,  &c> 

Many  of  the  tints  of  the  primary  colors,  and  many  of  the  hues  of 
the  secondaries  and  tertiaries,  have  also  their  popular  names  by 
which  they  are  recognized,  but  in  most  cases  it  would  be  very 
difficult  to  define  the  strength  of  the  tint,  or  composition  of  the 
hue  to  which  these  names  are  applied ;  light  tints  of  yellow  for 
example,  are  known  as  cream  color,  straw  color,  primrose,  &c., 
light  tints  of  red  are  known  as  rose-white,  carnation,  coquelicot, 
&e.;  and  of  blue,  as  pearl-white,  french  grays,  silver  grays  and 
azure. 

From  the  combination  of  yellow  with  red,  are  formed  gold  color, 
giraffe  and  scarlet;  from  a  mixture  of  red  and  blue,  result,  crim- 
son, pink,  rose  colors,  lakes,  peach-blossom,  lilac  and  violet;  and 
different  hues  of  green  are  known  as  emerald  green,  grass  green, 
invisible  green,  &c.  Numerous  other  names  might  be  added,  but 
at  best  they  contain  but  indefinite  ideas  of  the  tints  or  hues  meant, 
and  perhaps  no  two  artists  if  called  upon  to  compound  one  of  them, 
would  produce  the  same  result;  in  fact,  this  indefiniteness  of 
nomenclature  is  common  to  the  whole  scale  of  colors,  as  the  tints 
and  hues  so  run  into  each  other  by  almost  imperceptible  degrees, 
that  it  is  very  difficult  to  say  where  any  particular  hue  terminates 
and  the  next  begins. 

Another  element  of  confusion  in  the  popular  names  of  colors,  arises 
from  the  large  number  of  pigments  of  commerce  made  to  imitate 


134  THEORY     OF     COLOR. 

the  different  lines  and  tints  of  nature ;  each  bears  a  specific  name, 
either  derived  from  its  hue  or  from  the  elements  of  which  it  is 
composed :  but  as  they  are  made  by  different  manufacturers  of 
various  degrees  of  purity,  and  each  maker  has  his  own  standard  of 
hue  for  each  color,  we  need  not  hope  to  have  much  uniformity  of 
result. 

Before  closing  this  branch  of  our  subject,  we  will  recapitulate  in  a 
more  succinct  form,  some  of  the  facts  arrived  at ;  that  they  may 
make  more  permanent  impressions  on  the  memory. 

1st.  A  PRIMARY  COLOB  cannot  be  compounded  or  formed  by  any 
combination  of  the  other  colors  ;  they  are  three  in  number,  yellow,, 
red  and  blue. 

2nd.  A  SECONDARY  COLOR  is  formed  by  admixture  of  any  two  of  the 
primaries  in  equal  parts,  they  are  orange,  purple  and  green. 

3rd.  A  TERTIARY  COLOR  is  formed  by  admixture  of  .two  of  the  sec- 
ondaries, and  consequently,  contains  two  equivalents  of  one  of  the 
primaries,  and  one  of  each  of  the  others ;  these  are  also  three  in 
number,  citron,  russet  and  olive. 

4th.  BROKEN  COLORS  are  compounded  of  all  the  primaries  in  indefi- 
nite proportions,  and  form  three  classes,  brown,  maroon  and  gray. 

5th.  ADVANCING  COLORS  are  first,  those  which  advance  from  a  neu- 
tral point  toward  light ;  and  secondly,  those  which  cause  objects 
to  appear  more  prominent  in  a  composition,  and  consequently,  to 
appear  nearer  the  spectator. 

6th.  RETREATING  COLORS  are  first,  those  which  retreat  from  a  neu- 
tral point  toward  shade ;  and  secondly,  those  which  cause  objects 
to  appear  more  distant  or  to  recede  from  the  spectator. 

7th.  WARM  COLORS  are  those  in  which  red  or  orange  predominate ; 
they  are  generally  also  advancing  colors. 

8th.  COLD  COLORS  are  generally  also  retreating  colors,  in  which  bine 
and  neutral  gray  are  the  prevailing  hues. 

9th.  POSITIVE  COLORS  are  the  three  primaries,  and  their  definite 
compounds  as  placed  in  the  scale  at  page  124.  All  other  decided 
compounds  of  two  of  the  primaries  also  come  under  this  head. 

10th.  NEUTRALS  are  compounded  of  the  three  primaries  in  their 
neutralizing  proportions,  in  which  no  trace  of  the  positive  colors 
exist ;  they  are  neutral  black,  and  its  shades  neutral  gray. 

More  strictly  speaking,  we  have  theoretically  but  one  neutral ;  as 
the  neutral  grays  are  to  black,  what  the  lighter  tints  of  a  color 
are  to  its  full  tone,  viz.  the  same  color  diluted  ;  but  in  art,  the 


THEORY     OF     COLOR,  135 

pigments  of  white  and  black,  as  well  as  all  their  intermediate 
shades,  are  termed  neutral  colors. 

llth.  SEMI-NEUTRALS  are  the  same  as  the  broken  colors,  and  oc- 
cupy a  middle  space  between  the  positive  colors  and  the  neutral 
black. 

12th.  ARTIFICIAL  LIGHT  increases  the  brilliancy  of  all  warm  colors, 
but  it  deteriorates  all  hues  of  the  cold  colors,  and  entirely  neutral- 
izes many  of  their  lighter  tints. 


The  student  who  wishes  further  to  investigate  this  subject,  should 
refer  to  the  more  elaborate  treatises  which  have  been  published 
in  relation  to  it.  The  object  of  this  short  essay,  is  to  give,  in  a 
plain  and  simple  manner,  a  short  synopsis  of  the  science  for  the 
use  of  practical  men,  who  have  but  little  time  to  devote  to  its 
study.  Among  other  works,  "Field's  Chromatics,"  and  his 
"Grammar  of  Coloring,"  and  "Hay  on  the  Laws  of  Harmonious 
Coloring"  will  be  found  very  useful,  and  I  would  here  acknow- 
ledge my  indebtedness  to  them  for  many  of  the  facts  herein  con- 
tained. 

Numerous  small  books  of  instruction  in  the  art  of  coloring  and  of 
different  styles  of  painting  have  been  published  in  England,  and 
are  easily  available  here  at  moderate  cost ;  the  conclusion  of  this 
woik  will  therefore  be  devoted  to  the  coloring  of  architectural  and 
mechanical  drawings,  and  to  specify  some  of  the  most  useful 
pigments  for  the  purpose. 


136 


GEOMETRICAL  DRAWINGS. 


MECHANICAL  or  ARCHITECTURAL  DRAWINGS  consist  of  plans,,  sec- 
tions, elevations,  details,  isometrical  and  perspective  views;  these 
terms  have  already  been  explained  at  pages  60,.  61. 

Drawings  may  also  bo  further  divided  into  two  classes :  first,  the 
more  elaborately  finished  drawings  for  the  purpose  of  explaining 
the  proposed  construction  entire  and  in  all  its  parts,  for  the  use 
of  the  proprietor  and  contractor,  and  which  form  in  connection 
with  accompanying  specifications  the  basis  of  the  contract :  these 
should  be  made  as  full  and  explicit  as  possible,  to  prevent  the 
misconceptions  and  disputes  which  often  arise  from  the  insuffi- 
ciency of  those  important  requisites. 

The  second  class  consists  of 'outline  drawings,  showing  the  details 
of  construction,  generally  drawn  to  a  larger  scale  and  on  which 
the  dimensions  should  be  figured. 

Detail  Drawings  are  often  roughly  tinted  to  show  the  materials,  as 
well  as  the  forms  and  sizes  of"  construction;  in  machine  drawings 
where  round  bodies  frequently  occur,  they  are  shaded  by  ruling 
parallel  ink  lines,  almost  touching  each  other  for  the  darkest 
shades,  gradually  placing  them  wider  apart  as  they  approach  the 
light;  such  drawings  are  generally  roughly  made,  the  workman 
depending  more  on  the  figured  dimensions  than  on  the  scale^ 

For  detail  drawings,  a  plain  square  edged  board  and  any  strong 
paper  is  sufficient,  the  paper  can  be  secured  to  the  board  by  small 
flat  headed  drawing  pins  made  for  the  purpose. 

Plain  drawing  boards  are  frequently  made  with  clamps  firmly 
glued  and  screwed  to  the  ends,  or  battens  made  equally  secure  to 
the  back  of  the  board  :  both  these  methods  are  objectionable,  as 
all  wood,  however  well  seasoned,  will  contract  and  expand  with 
the  hygrometic  changes  of  the  atmosphere,  with  a  force  that  no 
clamps  can  restrain;  the  boards  will,,  consequently,  warp  or  split 
under  such  circumstances ;  a  better  method  of  constructing  them,, 
is  to  sink  dovetailed  grooves  in  the  back  of  the  board,  and  drive 
in  battens  to  fit  the  grooves,  these  require  no  fastening  and  will 
allow  the  contractions  or  expansions  to  take  place  freely,  and  at 


GEOMETEICAL    DRAWINGS.  137 

the  same  time  retain  the  board  straight  and  sound:  a  drawing 
board  large  enough  for  imperial  paper,  made  of  soft  pine  half  an 
inch  thick,  with  grooves  sunk  a  quarter  of  an  inch  deep,  the 
battens  four  inches  wide  by  three-quarters  thick,  has  been  in  use 
for  ten  years,  and  is  now  as  straight  and  free  from  cracks  as  when 
it  was  first  made. 

For  finely  finished  colored  drawings,  the  paper  should  be  dampened 
and  stretched  smoothly  on  the  board  with  its  edges  firmly  secured, 
otherwise,  the  parts  which  are  made  wet  with  the  color,  will  stretch 
and  rise  from  the  board,  and  as  a  natural  consequence,  the  color  will 
flow  toward  the  lowest  parts,  and  prevent  your  obtaining  an  even 
tint.  Drawing  boards  for  this  purpose  should  consist  of  a  rebated 
frame  with  a  thin  panel,  with  buttons  or  bars  on  the  back  to  press 
the  panel  tightly  against  the  frame.  Whatman's  rough  surface 
drawing  papers  are  best  for  this  purpose,  the  sizes  are  as  follows  : 

Cap,  17x14  in.         Elephant,  28x23  in. 

Demy,  18x15,  Imperial,  30x22, 

Medium,  22x17,  Columbier,  34x24, 

Royal,  24x19,  Atlas,  33x26, 

Super  Royal,     27x19,  Double  Elephant,  40x27. 

Antiquarian,  52x31  in. 

These  papers  have  a  water  mark  of  the  maker's  name,  and  the  side 
on  which  this  name  reads  correctly,  is  intended  as  the  right  side 
to  make  the  drawing  on.  To  stretch  the  paper,  wet  the  back  of 
the  sheet  evenly  and  lightly  with  fair  water,  either  with  a  flat 
brush  or  sponge,  or  by  laying  a  wet  cloth  on  it ;  in  this  state,  the 
sheet  will  continue  to  expand  in  size  until  the  moisture  has  pene- 
trated its  whole  substance,  which  requires  but  a  few  minutes  to 
effect ;  the  sheet  should  be  about  one-and-a-half  inches  larger  than 
the  panel  in  each  direction ;  it  should  then  be  placed  on  the  panel 
projecting  equally  all  around,  its  edges  pressed  down  into  the  re- 
bate, and  the  panel  secured  firmly  in  the  frame  which  clamps  the 
paper  and  prevents  its  contraction  when  drying,  so  that  the  sheet 
remains  permanently  stretched,  and  will  remain  smooth  and  even 
while  the  color  is  applied.  When  paper  is  stretched  on  a  plain 
board,  its  edges  are  secured  to  the  board  with  strong  paste ;  but  in 
this  case,  a  damp  cloth  should,  be  kept  on  it  until  the  paste  is 
sufficiently  dry,  to  prevent  its  being  drawn  off  by  the  too  rapid 
contraction  of  the  sheet ;  as  soon  as  the  paper  is  dry,  it  is  ready 
for  the  drawing. 

18 


138  GEOMETEICAL    DRAWINGS. 

The  other  requisites  for  geometrical  drawing,  are  a  T  square,  a  pair 
of  wooden  right  angled  triangles,  the  one  with  equal  &ides  and 
angles  of  45  degrees ;  the  other  with  angles  of  30  and  60  degrees 
for  isometrical  drawing ;  pencils,  rubber,  a  case  of  drawing  instru- 
ments, and  for  drawing  ink  lines,  a  cake  of  India  ink,  aporcelain 
slab  for  rubbing  the  ink  on,  and  camel's  hair  pencils  for  applying 
the  ink  to  the  drawing  pens. 

The  construction  and  use  of  the  scales  and  parallel  ruler,  have 
already  been  explained  in  the  early  part  of  the  Drawing  Book  ; 
the  other  instruments  in  a  case,  require  but  little  explanation, 
their  uses  being -almost  self-evident  to  any  intelligent  student. 
Pencils  should  be  of  good  quality,  sufficiently  hard  to  retain  a  good 
point,  but  not  so  hard  as  to  cut  the  paper — and  it  is  a  very  impor- 
tant requisite  in  a  pencil  that  its  marks  may  be  easily  erased ;  it 
is  almost  impossible,  to  point  a  pencil  properly  with  a  dull  knife, 
you  should  therefore,  keep  a  keen  one,  it  will  save  your  time,  pen- 
cils and  temper. 

Common  writing  ink  should  never  be  used  in  the  drawing  pens,  it 
would  soon  eat  away  the  metal  and  render  them  useless.  India 
ink  rubbed  up  with  pure  water,  is  the  best  for  the  purpose,  as  it 
can  be  made  of  any  required  shade ;  a  slight  addition  of  carmine 
will  cause  the  ink  to  flow  more  freely ;  the  pen  should  be  set  to 
the  requisite  degree  of  fineness,  and  need  not  be  altered  to  add  a 
fresh  supply  of  ink,  this  should  be  applied  to  the  side  of  the  pen 
with  a  hair-pencil,  which  should  be  occasionally  passsed  between 
the  blades  of  the  pen  to  prevent  its  clogging,  any  surplus  ink  on 
the  outside  of  the  blades  should  be  wiped  off  before  applying  it  to 
the  paper.  In  using  the  drawing  pen  either  for  straight  or  circu- 
lar lines,  it  is  very  important  to  have  both  the  blades  to  rest  on 
the  paper,  or  the  lines  will  be  irregular. 

With  the  f  square  applied  to  the  edges  of  the  board  all  lines  paral- 
lel to  any  of  its  sides  may  be  drawn  correctly ;  parallel  inclined 
lines  may  be  drawn  by  a  square  having  one  of  its  sides  moveable, 
and  clamped  by  a  screw  to  the  required  angle,  or  with  the  parallel 
ruler,  or  by  moving  a  triangle  along  a  straight  edge.  The  outlines 
of  the  drawing  should  be  first  made  in  pencil,  then  the  permanent 
lines  ruled  in  India  ink  of  a  light  shade,  and  the  surplus  pencil 
marks  removed  with  the  rubber,  or  with  a  crumb  of  bread,  which 
will  answer  the  same  purpose ;  the  drawing  is  then  ready  for  col- 
oring. 


139 


APPLICATION  OF  COLORS. 


THE  student  who  has  carefully  read  the  first  part  of  this  essay,  will 
readily  suppose  that  with  three  good  pigments  representing  the 
three  primary  colors,  a  white  for  lights  and  a  black  for  shades,  he 
can  mix  any  tint  or  hue  he  may  require  ;  and  this  is  true  to  a 
considerable  extent,  but  not  altogether  so,  as  unfortunately,  colored 
pigments  are  not  always  homogeneous  in  their  composition,  and 
often  act  chemically  upon  each  other,  so  as  to  change,  or  perhaps 
neutralize  the  original  hues ;  the  knowledge  of  the  chemical  prop- 
erties of  colors  requires  more  study  and  attention  than  is  necessary 
for  the  mechanical  draftsman  to  devote  to  it,  as  the  colorman  has 
prepared  numerous  pigments  of  different  hues  to  meet  his  wants  ; 
these  pigments  are  commonly  called  colors,  and  in  this  practical 
branch  of  our  subject,  we  shall  follow  the  common  practice  and 
use  the  terms  synonymously. 

Colors  are  either  transparent  or  opaque,  a  transparent  color  is  often 
laid  over  another  color  to  change  or  modify  its  hue,  and  will  often 
produce  a  much  more  soft  and  agreeable  tone  than  could  be  pro- 
duced by  a  single  color  ;  this  operation  is  technically  called  glazing. 
The  draftsman  will  often  find  occasion  to  exercise  his  ingenuity  in 
this  respect  as  well  as  in  the  mixing  of  his  hues,  and  on  this  sub- 
ject he  must  depend  chiefly  on  his  own  taste  and  judgment,  as  it 
is  very  difficult  to  give  any  other  than  general  rules  for  the  pur- 
pose. 

The  following  are  the  most  useful  of  the  prepared  cake  colors  for 
architectural  or  mechanical  drawings  : 
YELLOWS.     Gamboge, 

Roman  Ochre  or  Yellow  Ochre, 

Indian  Yellow. 
REDS.    .       Carmine  or  Crimson  Lake, 

Vermilion, 

Indian  Red. 
BLUES.          Cobalt  or  Ultra-marine, 

Prussian  Blue, 

Indigo. 
BROWNS.       Sepia, 

Vandyke  Brown  or  Burnt  Umber, 

Raw  Sienna. 


140  APPLICATION    OF    COLOES. 

India  ink  may  be  used  for  the  shades  or  a  semi-neutral  tint,  corn- 
pounded  of  indigo  and  Indian  red  ;  this  tint  may  be  found  in  cakes 
under  the  name  of  neutral  tint. 

The  surface  of  the  paper  should  represent  the  lights  in  water  color 
drawings,  or,  where  very  small  lights  are  required,  they  may  be 
scratched  out  with  the  point  of  a  knife  ;  the  last  method  is  gener- 
ally resorted  to  where  drawings  are  made  on  surface  tinted  or 
graduated  paper. 

The  secondary  colors,  orange,  green  and  purple,  as  well  as  the 
semi-neutral  grays  and  other  broken  colors,  may  all  be  compounded 
from  the  pigments  in  the  above  selection. 

A  little  practice  and  observation  of  the  results  of  the  different 
mixtures,  will  soon  enable  the  student  to  combine  them  satis- 
factorily ;  he  will  find  that  many  of  the  combinations  he  produces, 
will  result  in  a  foul,  cloudy  mixture,  these  he  should  note  par- 
ticularly, and  avoid  in  future ;  and  those  which  result  satis- 
factorily, should  be  still  more  carefully  remembered  for  future 
use.  As  a  general  rule,  which  should  be  well  fixed  in  the 
memory,  the  fewer  the  pigments  used  in  the  composition  of  a  tint, 
the  more  clear  and  satisfactory  will  be  the  hue  produced. 

In  using  any  hue  compounded  of  different  colors,  it  should  be  con- 
stantly stirred  up  with  a  brush  whilst  taking  a  fresh  supply  in  it, 
as  such  mixtures  have  always  a  tendency  to  separate,  this  must 
be  more  particularly  attended  to  when  vermilion  enters  the  com- 
pound, as  it  is  so  much  heavier  than  other  pigments  that  it  in- 
variably falls  to  the  bottom. 

In  mechanical  drawings,  the  plans  and  sections  especially,  should  be 
so  colored  as  to  show  the  materials  and  construction ;  so  that  this 
information  may  be  obtained  by  the  mere  inspection  of  the  draw- 
ings, without  the  trouble  of  hunting  it  up  in  the  specifications, 
which  are  seldom  available  at  the  place  of  construction,  being  gen- 
erally filed  away  in  the  office  for  particular  reference.  In  shading 
such  drawings,  the  shadows  should  always  be  projected  at  an  angle 
of  45  degrees,  as  explained  at  page  113,  where  shadows  are 
especially  treated  of. 

In  elevations,  it  is  not  so  important  that  the  materials  should  be  so 
palpably  designated  ;  but  even  here  it  should  not  be  altogether  ne- 
glected, more  particularly  when  the  materials  are  to  remain  of 
their  natural  color,  as  the  general  coloring  has  often  an  important 
effect  on  the  character  of  the  design. 


APPLICATION   OF   COLOES.  141 

In  perspective  views,  this  requisite  is  generally  lost  sight  of,  and 
often  in  architectural  views,  the  coloring  and  shading  are  more  in 
accordance  with  the  practice  of  the  landscape  painter,  than  with 
that  of  the  mechanical  draughtsman. 

The  colors  used  in  plans  and  sections  to  denote  the  construction 
should  be  as  nearly  as  possible  the  colors  of  the  materials  to  be 
employed.  A  very  excellent  article  on  this  subject  has  recently 
been  published  in  an  English  translation  of  a  popular  French  work 
on  drawing;  "The  Practical  Draughtsman."  As  this  most  likely 
embodies  the  practice  adopted  in  both  those  countries,  it  is 
desirable  that  we  in  America,  should  assimilate  our  practice  to 
theirs,  so  that  every  one  may  know  the  materials  intended  to  be 
used  in  a  construction,  by  an  inspection  of  the  tints,  no  matter  where 
the  drawing  may  have  been  made. 

The  following  extracts  have  been  taken  from  this  work,  I  would 
recommend  their  careful  study,  as  with  the  few  additions  given 
below,  they  pretty  well  cover  the  whole  ground.  The  colors 
designated,  are  termed  "conventional  colors,  that  is,  certain  colors 
are  generally  understood  to  indicate  particular  materials."  India 
ink  in  these  extracts,  is  called  China  ink,  either  name  may  be  ap- 
plied to  this  pigment. 

''Stone. — This  material  is  represented  by  a  light  dull  yellow,  which 
is  obtained  from  Roman  ochre,  with  a  trifling  addition  of  China 
ink." 

This  color  does  very  well  for  light  sand-stone,  but  where  different 
kinds  of  stone  are  to  be  used  in  a  construction,  other  hues  are 
necessary  to  denote  them. 

Granite  may  be  indicated  by  Prussian  blue,  with  a  little  India  ink 
added. 

Red  sand-stone,  by  Indian  red. 

White  or  other  light  colored  marbles,  by  a  light  tint  of  yellow  ochre 
or  raw  sienna. 

Rubble  stone  walls,  by  Prussian  blue  with  Indian  red ;  these  should 
be  further  designated  in  sections,  by  a  few  irregular  lines  of  a  differ- 
ent tint  or  hue,  laid  in  with  the  hair  pencil  after  the  first  tint  is 
dry,  to  represent  the  joints  and  indicate  the  construction. 

"Brick. — A  light  red  is  employed  for  this  material,  and  may  be 
obtained  from  vermilion,  which  may  sometimes  be  brightened  by 
the  addition  of  a  little  carmine.  A  pigment  found  in  most  color- 
boxes  and  termed  light  red,  may  also  be  used  when  great  purity 


142  APPLICATION    OF    COLORS. 

and  brightness  of  tint  is  not  wanted.  If  it  is  desired  to  distinguish 
firebrick  from*  the  ordinary  kind,  since  the  former  is  lighter  in 
color  and  inclined  to  yellow,  some  gamboge  must  be  mixed  with 
the  vermilion,  the  whole  being  laid  on  more  faintly. 

"/Steel  or  Wrought  Iron. — The  color  by  which  these  metals  are  ex- 
pressed is  obtained  from  pure  Prussian  blue  laid  on  light :  being 
lighter  and  perhaps  brighter  for  steel  than  for  wrought-iron. 

"Cast-Iron. — Indigo  is  the  color  employed  for  this  metal;  the  ad- 
dition of  a  little  carmine  improves  it.  The  colors  termed  Neutral 
Tint  or  Paynes  Gray,  are  frequently  used  in  place  of  the  above, 
and  need  no  further  admixture.  They  are  not,  however,  BO  easy  to 
work  with,  and  do  not  produce  so  equable  a  tint. 

" Lead  and  Tin  are  represented  by  similar  means,  the  color  being 
rendered  more  dull  and  gray  by  the  addition  of  China  ink  and  car- 
mine or  lake. 

"Copper. — For  this  metal,  pure  carmine  or  crimson  lake  is  proper. 
A  more  exact  imitation  of  the  reality  may  be  obtained  by  the 
mixture  with  either  of  these  colors,  of  a  little  China  ink  or  burnt 
sienna — the  carmine  or  lake,  of  course,  considerably  predominating. 

"Brass  or  Bronze. — These  are  expressed  by  an  orange  color,  the 
former  being  the  brighter  of  the  two ;  burnt  Eoman  ochre  is  the 
simplest  pigment  for  producing  this  color.  Where,  however,  a 
very  bright  tint  is  desired,  a  mixture  should  be  made  of  gamboge 
with  a  little  vermilion — care  being  taken  to  keep  it  constantly 
agitated  as  before  recommended.  Many  draughtsmen  use  simple 
gamboge  or  other  yellow. 

"  Wood. — It  will  be  observable  from  preceding  examples,  that  the 
tints  have  been  chosen  with  reference  to  the  actual  colors  of  the 
materials  which  they  are  intended  to  express — carrying  out  the 
same  principle,  we  should  have  a  very  wide  range  in  the  case  of 
wood.  The  color  generally  used,  however,  is  burnt  umber  or  raw 
sienna ;  but  the  depth  or  strength  with  which  it  is  laid  on,  may 
be  considerably  varied.  It  is  usual  to  apply  a  light  tint  first,  sub- 
sequently showing  the  graining  with  a  darker  tint,  or  perhaps  with 
burnt  sienna.  These  points  are  susceptible  of  great  variation,  and 
very  much  must  be  left  to  the  judgment  of  the  artist. 

"Leather,  Vulcanized  India  Rubber  and  Gutta  Percha. — These 
are  all  represented  by  very  similar  tints.  Leather  by  light,  and 
gutta  percha  by  dark  sepia,  whilst  vulcanized  India  rubber  requires 
the  addition  of  a  little  indigo  to  that  color. 


APPLICATION   OF    COLOKS.  143 

"Manipulation  of  the  Colors. — The  cake  of  color  should  never  be 
dipped  in  the  water,  as  this  causes  the  edges  to  crack  and  crumble 
off,  wasting  considerable  quantities.  Instead  of  this,  a  few  drops 
of  water  should  be  first  put  in  the  saucer  or  on  the  plate,  and  then 
the  required  quantity  of  color  rubbed  down,  the  cake  being  wetted 
as  little  as  is  absolutely  necessary.  The  strength  or  depth  of  the 
color  is  obtained  by  proportioning  the  quantity  of  water,  the  whole 
being  well  mixed,  to  make  the  tint  equable  throughout.  When 
large  surfaces  have  to  be  covered  by  one  tint,  which  it  is  desired 
to  make  a  perfectly  even,  flat  tint,  it  is  well  to  produce  the  re- 
quired strength  by  a  repetition  of  very  light  washes.  These 
washes  correct  each  other's  defects,  and  altogether  produce  a  soft 
and  pleasing  effect.  This  method  should  generally  be  employed 
by  the  beginner,  as  he  will  thereby  more  rapidly  obtain  the  art  of 
producing  equable  flat  tints.  The  washes  should  not  be  applied 
before  each  preceding  one  is  perfectly  dry.  When  the  drawing- 
paper  is  old,  partially  glazed,  or  does  not  take  the  color  well  its 
whole  surface  should  receive  a  wash  of  water,  in  which  a  very 
small  quantity  of  gum-arabic  or  alum  has  been  dissolved.  In  pro- 
ceeding to  lay  on  the  color,  care  should  be  taken  not  to  fill  the 
brush  too  full,  whilst,  at  the  same  time,  it  must  be  replenished 
before  its  contents  are  nearly  expended,  to  avoid  the  difference  in 
tint  which  would  otherwise  result.  It  is  also  necessary  first  to  try 
the  color  on  a  separate  piece  of  paper,  to  be  sure  that  it  will  pro- 
duce the  desired  effect.  It  is  a  very  common  habit  with  water- 
color  artists  to  point  the  brush,  and  take  off  any  superfluous  color, 
by  passing  it  between  their  lips.  This  is  a  very  bad  and  disagree- 
able habit,  and  should  be  altogether  shunned.  Not  only  may  the 
color  which  is  thus  taken  into  the  mouth  be  injurious  to  health, 
but  it  is  impossible,  if  this  is  done,  to  produce  a  fine  even  surface, 
for  the  least  quantity  of  saliva  which  may  be  taken  up  by  the 
brush  has  the  effect  of  clouding  and  altogether  spoiling  the  wash 
of  color  on  the  paper.  In  place  of  this  uncleanly  method,  the 
artist  should  have  a  piece  of  blotting-paper  at  his  side — the  more 
absorbent  the  better.  By  passing  the  brush  over  this  any  super- 
fluous color  may  be  taken  off,  and  as  fine  a  point  obtained  as  by 
any  other  means.  The  brush  should  not  be  passed  more  than 
once,  if  possible,  over  the  same  part  of  the  drawing  before  it  is 
dry ;  and  when  the  termination  of  a  large  space  is  nearly  reached, 
the  brush  should  be  almost  entirely  freed  from  the  color,  otherwise, 


144  APPLICATION    OF    COLOES. 

the  tint  will  be  left  darker  at  that  part.  Care  should  be  taken  to 
keep  exactly  to,the  outline;  and  any  space  contained  within  defi- 
nite outlines  should  be  wholly  covered  at  one  operation,  for  if  a 
portion  is  done,  and  then  allowed  to  dry,  or  become  aged,  it  will 
be  almost  impossible  to  complete  the  work,  without  leaving  a  dis- 
tinct mark  at  the  junction  of  the  two  portions.  Finally,  to  produce 
a  regular  and  even  appearance,  the  brush  should  not  be  over- 
charged, and  the  color  should  be  laid  on  as  thin  as  possible ;  for 
the  time  employed  in  more  frequently  replenishing  the  brush, 
because  of  its  becoming  sooner  exhausted,  will  be  amply  repaid  by 
the  better  result  of  the  work  under  the  artist's  hands." 

With  this  information  carefully  digested,  in  addition  to  the  in- 
structions of  much  the  same  tenor,  previously  given  for  laying  in 
shades  and  shadows,  at  page  116,  the  student  will  find  but  little 
difficulty  in  coloring  his  drawings  creditably  ;  but  he  will  require 
practice,  that  he  may  gain  experience  in  the  manipulation  of  hues 
and  tints,  and  freedom  in  the  use  of  the  brush. 

It  is  important  that  he  should  first  practice  on  drawings  that  re- 
quire but  little  labor  to  construct;  so  that  the  fear  of  spoiling 
them  would  not  cramp  his  motions  and  prevent  his  acquiring  the 
requisite  freedom  of  hand ;  if  a  drawing  of  this  character  be 
spoiled,  it  will  be  of  very  little  consequence,  as  it  can  be  easily 
replaced ;  with  an  elaborate  drawing  the  case  is  very  different, 
the  fear  of  spoiling  it  will  often  induce  hesitation  and  awkward- 
ness, and  produce  the  result  feared. 

When  a  drawing  is  to  be  shaded  and  tinted,  the  shades  and 
shadows  should  be  laid  in  with  India  ink  before  applying  the 
colors,  but  not  so  dark  as  required  for  the  finished  drawing,  as 
they  can  be  worked  up  to  better  advantage  by  deepening  them 
with  the  local  hues ;  sometimes  drawings  are  shaded  with  differ- 
ent tints  of  the  local  colors  without  first  shading  with  ink  ;  the 
student  should  practice  in  both  styles,  so  that  he  may  be  able  to 
apply  either  or  mix  them,  as  his  taste  and  judgment  may  direct. 

In  shading  perspective  views  of  exteriors,  the  student  must  re- 
member that  all  his  shades  and  hues  must  diminish  in  intensity 
as  the  distance  increases ;  but  in  limited  interior  views,  as  of  a 
room,  his  shades  will  increase  in  depth  toward  the  rear  of  the 
picture,  unless  such  room  is  lighted  by  side  windows  or  by  other 
light  than  that  obtained  from  the  front  of  the  picture  where  the 
section  of  the  room  is  made. 


TO   MOUNT   DEAWINGS,  ETC.  145 

The  student  who  endeavors  to  follow  those  instructions,  will 
doubtlessly  meet  with  some  difficulties,  but  he  will  soon  overcome 
them,  if  he  will  practice  with  the  requisite  attention  and  industry. 

Every  obstacle  encountered  should  be  removed  from  the  path  at 
once,  he  must  not  pass  over  or  around  it,  leaving  it  as  a  stumbling 
block  to  his  future  labors ;  but  should  attack  each  in  detail,  and 
let  every  encounter  act  as  a  spur  to  further  progress  and  to  ulti- 
mate success. 


TO  MOUNT  DRAWINGS. 

It  is  often  desirable  for  large  drawings  that  are  expected  to  be  much  handled, 
or  to  be  preserved  for  records,  to  strengthen  them  by  backing  with  cotton 
cloth,  technically  called  "mounting"  this  may  be  done  as  follows : 

1st.  Procure  a  piece  of  new  cotton  cloth  several  inches  larger  each  way  than  the 
drawing,  and  tack  its  edges  securely  down  on  a  table,  drawing-board  or  other 
flat  surface  ;  new  cloth  is  preferred  because  it  shrinks  considerably  in  drying ; 
that  which  has  been  washed  has  lost  this  property  of  shrinkage,  having  already 
gone  through  the  process. 

2nd.  Prepare  a  strong  flour  paste,  about  the  consistency  of  that  used  by  paper- 
hangers,  and  with  a  large  brush  give  the  back  of  the  drawing  a  coat  of  the 
paste  ;  be  careful  to  cover  the  whole  of  the  paper,  that  it  may  stretch  equally. 

3rd.  Coat  the  cloth  thoroughly,  so  that  it  may  be  saturated  with  paste. 

4th.  Lay  the  drawing  in  position  on  the  cloth,  then  with  two  stout  bone  or 
ivory  paper  folders,  commence  at  the  middle  of  the  sheet  with  the  edges  of 
the  folders  to  press  in  every  direction  from  the  centre  toward  the  edges,  until 
all  the  surplus  paste  is  pressed  out ;  be  careful  to  keep  the  face  of  the  draw- 
ing dry  and  clean  and  also  the  edges  of  the  folders,  otherwise  the  surface  of 
the  paper  may  be  injured,  more  especially  if  it  has  much  color  on  it.  When 
the  whole  is  dry,  trim  the  edges  and  bind  them  with  a  narrow  ribbon. 

When  a  drawing  requiring  heavy  tints  of  color  is  intended  to  be  backed,  in 
order  to  avoid  the  risk  of  smearing  in  the  process,  it  would  be  better  to  omit 
those  tints  until  afterward ;  they  should  be  laid  in  after  the  paper  is  dry  and 
before  cutting  it  loose  from  the  table,  when  a  satisfactory  result  may  be 
arrived  at. 


TO  CLEAN  DRAWINGS  OR  ENGRAVINGS. 

A  draughtsman  occasionally  desires  to  save  a  dilapidated  drawing  or  old  en- 
graving by  backing.it,  this  may  be  done  as  above  directed,  but  if  it  is  much 
soiled  by  smoke  or  otherwise,  it  is  desirable  to  expel  the  stains  as  much  as 
possible  before  mounting.  I  have  found  the  following  process  satisfactory  : 

1st.  Place  the  engraving  or  drawing  face  downward  in  a  pan  of  clear  cold 
water,  let  it  soak  some  hours,  more  or  less,  depending  on  the  strength  of  the 

19 


146  TRANSPARENT    TRACING   CLOTH,  ETC. 

paper,  the  longer  it  remains  in  the  water  without  injuring  the  texture  of  the 
paper,  the  cleaner  it  will  become. 

2nd.  Remove  the  sheet  carefully  from  the  water  and  lay  it  face  downward  on 
a  clean  flat  table  or  drawing-board  ;  then  with  a  soft  sponge  absorb  the 
moisture,  adding  clean  water  from  time  to  time,  until  all  the  removable  stains 
have  been  extracted ;  then  turn  the  sheet  over  and  proceed  very  carefully  in 
the  same  way  with  the  face ;  the  sponge  must  not  be  rubbed  on  the  paper,  but 
pressed  on  and  wrung  out  until  the  water  thus  absorbed  becomes  clear. 

3rd.  Dry  the  drawing  between  sheets  of  thick  blotting  paper  under  pressure, 
when  it  will  be  ready  for  mounting.  This  process  will  remove  most  of  the 
stains  caused  by  water  or  smoke,  but  will  not  affect  those  made  by  oil  or 
grease,  these  should  be  removed  before  wetting  the  print,  by  placing  blotting 
paper  over  the  stain  and  extracting  the  grease  by  pressing  the  paper  with  a 
hot  smoothing  iron. 

The  printer's  ink  of  an  engraving  is  insoluble  in  water  and  not  likely  to 
take  injury  from  this  method ;  the  same  may  be  said  of  a  drawing  made  with 
India  ink,  but  if  it  has  been  made  with  common  ink,  the  drawing  can  only  be 
cleaned  by  rubbing  with  soft  rubber,  or  with  crumb  of  stale  bread. 


TRANSPARENT  TRACING  CLOTH. 

Is  now  much  used  because  of  its  greater  strength,  in  preference  to  tracing  paper  for 
taking  copies  of  drawings  for  the  workman  or  for  preservation ;  this  cloth  is 
liable  to  injury  from  dampness,  and  being  transparent  requires  to  be  placed 
on  a  board  or  paper  for  examination.  If  the  tracings  and  letterings  are  drawn 
with  India  ink  of  good  quality,  the  whole,  when  the  ink  has  become  dry,  may  be 
dipped  in  water,  which  extracts  the  transparent  preparation ;  then  dried  and 
pressed  with  a  hot  iron,  when  you  will  have  a  permanent  drawing  on  an 
opaque  surface,  but  little  liable  to  injury  from  moisture. 


QUALITY  OF  MATERIALS. 

The  draughtsman  who  values  his  time,  will  always  find  it  the  more  economical 
plan,  to  procure  all  his  materials  of  the  best  quality,  and  also  to  keep  his  in- 
struments clean  and  in  good  condition  ;  he  will  thus  be  enabled  with  the  same 
degree  of  effort  to  produce  a  much  more  satisfactory  drawing. 


INDEX. 


PAG3 


Abscissa,       ......  .4 

Absorption  of  light,         ....  .82 

Accidental  points,     .....  .         8J 

Aerial  perspective,          .... 

Altitude  of  a  triangle,  ....  .         1( 

Angles  described,  .... 

Angle  of  incidence,  ....  .8] 

"      Visual  ....  .84 

"      How  to  draw  angles  of  45°,  .. 
Apex  of  a  pyramid,         ....  .39 

"     of  a  cone,        .....  .44 

Application  of  the  rule  of  3,  4  and  5,  17 

Apparent  size  of  an  object,   ....  .90 

Architrave,          .......  66 

Arc  of  a  circle,         .....  .1 

Arcades  in  perspective,  .....  9 

Arches — Composition  of       ....  .54 

Construction  of  .....  54 

"        Definitions  of          ....  .54 

Arch — Thrust  of  an         .  .  .  .  .  .  55 

"       Amount  of  the  thrust  of  an  arch,  (note)  .  .  .55 

"       Straight  arch  or  plat  band,  ....  55 

"       Rampant       .......         55 

"       Simple  and  complex  arches,         ....  55 

"       Names  of  arches,       .  .  .  .  .  .55 

Arches  in  perspective,    .  .  .  .  .  .    98  to  100 

Arithmetical  perspective,  .  .  .  .  .83 

Aspect  of  a  country  house,  .          ;.».*         .  .  .  61 

Axis  of  a  pyramid,  •  .  .  .  .  .39 

"    of  a  sphere,  ......  41 

"    of  a  cylinder,  ......         42 

"    Major  and  minor  axes,        .  .  .  .  43, 44 

"    of  a  cone,         .......        45 

"    Difference  between  the  length  of  the  axes  in  the  sections  of  the  cone 

and  cylinder,  ......         46 

"    of  tne  parabola,      ......  50 

Axioms  in  perspective,          ......         90 

Back  of  an  arch  or  Extrados,      .....  55 

Band,  listel  or  fillet,  ......         68 

Base  of  a  triangle,         .  .  ; 


OF  THR 


148 


PAGE. 


Base  of  a  pyramid,  .......         39 

"    of  a  cone,               ......  44 

"    of  a  Doric  column,       ......         68 

"    line  or  ground  line,            .             .             .             .             .  86 

Bead  described,        .......         68 

Bed  of  an  arch,               ......  55 

Bisect — To  bisect  a  right  line,           .             .             .             .  .14 

"         To  bisect  an  angle,         .             .             .             .             .  18 

Bird's  eye  view,         ...             .             .             .  .89 

Cavetto,  a  Roman  moulding,         .             .             .             .             .  69 

Centre  of  a  circle,      .  .           .             .             .             .             .  .11 

"      of  a  sphere,           ......  41 

"      of  a  vanishing  plane,                .             .             ,             .  .110 

Chords  defined,                .             .             .                           .             .  12 

"      Scale  of  chords  constructed,  .             .             .             .  .26 

"      Application  of  the  scale  of  chords,            ...  26 

Circle  described,        .             .             .             .             .             .  .11 

•  "      To  find  the  centre  of  a  circle,                                    .  22 

"      To  draw  a  circle  through  three  given  points,  .             .  .23 

"      To  find  the  centre  for  describing  a  flat  segment,     .             .  23 

"      To  find  a  right  line  equal  to  a  semicircle,         .             .  .23 

"        "        "         equal  to  an  arc  of  a  circle,        .             .  23 

"      Workmen's  method  of  doing  the  same,             .             .  .24 

"      Great  circle  of  a  sphere,                  ....  41 

"      Lesser  circle  of  a  sphere,         .             .             .             .  .41 

"      Circumferences  of  circles  directly  as  the  diameters,              .  74 
"      in  perspective,             ......         95 

"     Application  of  the  circle  in  perspective,     ...  101 

Circular  plan  and  elevation,                 .             .             .             .  .67 

"       objects — Shading  of                     .             .             .             .  112 

Circumference  of  a  circle,       .             .             .             .             .  .11 

"           directly  as  its  diameter,               .             .  74 

Circular  domes — To  draw  the  covering  of     .             .             .  .52 

Color — How  to  color  shadows,  &c.          .             .             .             .  116 

Conjugate  axis  or  diameter,                 .             .             .             .  .  43,  44 

of  a  diameter  of  the  ellipsis,    ....  44 

Contents  of  a  triangle,           .             .             .             .             .  .10 

of  a  cube,         ......  37 

"       of  the  surface  of  a  cube,      .             .             .             .  .38 

Complement  of  an  arc  or  angle,                 .             .             .             .  13 

Complex  and  simple  arches,                .             .             .             .  .55 

Cone,  right,  oblique  and  scalene,               ....  40 

To  draw  the  covering  of  a  cone,            .             .             .  .40 

"    Sections  of  the  cone,           .  45 

Co-sine, ^         13 

Co-tangent,           .              .              ...              .              .             .  13 


149 

PAGE. 

Co-secant ,     .  ,  .  .  .  .  .  .13 

Construct — To  construct  a  triangle,         .  .  .  .  17 

"  an  angle  equal  to  a  given  angle,      .  .         18 

"  "  an  equilateral  triangle  on  a  given  line,  .  19 

"  "  a  square  on  a  given  line,     .  .  .20 

"  "  a  pentagon  on  a  given  line,      .  .  21 

"  a  heptagon  on  a  given  line,  .  .         22 

"  "  any  polygon  on  a  given  line,    .  .  22 

"  "  a  scale  of  chords    ....         26 

"  "  the  protractor,  ...  30 

Construction  of  arches,          ......         54 

Contrary  flexure  — Curve  of  .       »     .  .  .  7 

«          «         Arch  of   : 57 

Cornice  and  piazza — Effect  of  the  ...  62 

Cottage  in  perspective,          .  .  .  .  .  .106 

Cohering  of  the  cube,  .  .  38 

"          "      parallelopipedon,      .  .  .  .  .38 

"  "      triangular  prism,  ....  38 

"          "      square  pyramid,        ....  39 

"          "      hexagonal  pyramid,        ....  40 

"          «      cylinder,      .  .  .  .  .  .40 

«          "      cone,     ......  40 

«          "      sphere, .41 

"          "      regular  polyhedrons,       ....  42 

"          "      circular  domes,         ...  .  .  .52 

Crown  of  an  arch,  .....  55 

"      moulding,      .......         69 

Cube  or  hexahedron,       .  .  .  .  .  .  37 

Cubic  measure,         .......         37 

Cube — To  draw  the  Isometricai  ....  76 

"      in  perspective,  .  .  .  .  .  .104 

Cycloid  described,  .*....  35 

Cycloidal  arches,       .......         36 

Cylinder,  •          .  •  •  •  *  •  •  40 

"       To  draw  the  covering  of  a   .  .  .  .  .40 

"       Sections  of  the  .  .  .  .  .42,43 

"       Right  and  oblique     ......         42 

u       To  find  the  section  of  a  segment  of  a  cylinder  through  three 

given  points,         ......         51 

"       of  a  locomotive  engine,  ....  75 

"       in  perspective,          .  .  .  .  .  .       102 

"       Shadow  of  a      .  .  .  .  .  116 

Cyme.  :r  cyma  recta — Roman,  •  .  •  •  .69 

"  «  Grecian,     .....  71 

"     reversa,  talon  or  ogee — Roman,  .         69 

«         c<  "          "         Grecian,    :  .  71 


150 


PAGE. 


Degree  defined,         .  .  .  •  •  •  .12 

Depressed  arch,  ...»••  59 

Design  for  a  cottage,  ...••• 

"      What  constitutes  a  .  .  .  *   -       '  «  61 

Details  of  cottage,   ....•••         65 

Diamond  defined,  ..»*•» 

Diagonal  defined,      ..••••• 

Diagonal  lines  in  perspective,      ..... 

Diameter  of  a  circle,  ..... 

"        of  a  sphere,       .  •  •  •  •  •  41 

"        of  an  ellipsis,          .  .  .  •  *  .  43,  44 

"        of  the  parabola,  r.  .  .  ••  -  60 

Definitions  of  lines,  *  •  •  ••  •  -  7 

"         of  angles,       .  .  .  •  »  * 

"         of  superficies,      .  .  ...  9 

"         of  the  circle,  .....  11 

u        of  solids,  ......         36 

"         of  the  cylinder,  .  .  .  .  .  42 

"         of  the  cone,          .  .  .  .  .  •         44 

"         of  the  parabola,  .  .  .  .  .  50 

"         of  arches,  .  .  .  »  .  .55 

"         in  perspective,  .....  86 

Directrix  of  the  parabola,      ......         50 

Distance — Points  of        ......  87 

«          Half 92 

"          The  quantity  of  reflected  light  enables  us  to  judge  of  .  Ill 

Dodecahedron,  .  .  .  .  •  •  •  .42 

Domes — Covering  of  hemispherical         ....  52 

Double  shadows,  .  .  •  •  •  .112 

Echinus,  or  Grecian  ovolo,          .  .  .  .  70 

Effect — A  perspective  view  necessary  to  shew  the  effect  of  an  intended 
improvement,  .  .  •  •  •  •  61 

Elevation  described,  ....  .  .         61 

Ellipsis — False  ......  34 

"         the  section  of  a  cylinder,   .  .  .  .  .43 

"         To  describe  an  ellipsis  with  a  string    ...  43 

"         the  section  of  a  cone,         .....         45 

"         To  describe  an  ellipsis  from  the  cone    ...  45 

"         To  describe  an  ellipsis  by  intersections       .  .  .46 

"         To  describe  an  ellipsis  with  a  trammel  .  .  47 

Elliptic  Arch — To  draw  the  joints  of  an         .  .  .  .57 

Epicycloid  described,      ......  36 

Equilateral  triangle,  ......  9 

"          arch,  (Gothic)  .....  58 

Extrados  or  back  of  an  arch,  .  .  .  .  .55 

Fillet,  band  or  listel,       ......  68 


151 


Focus — Foci  of  an  ellipsis,                                           ..  ,43 

"        of  a  parabola,     .  50 

Foreshortening,         .  .  .  ...         84 

"  The  degree  of  foreshortening  depends  on  the  angle  at 

which  objects  are  viewed,  ....  85,  90 

Form  of  shadows,           .             .             .             •                          .  Ill 

Frustrum  of  a  pyramid,       •  .           •  .             .             .  .39 

of  a  cone,        .             .             .             .             .             .  45 

Globe  or  sphere,        .             .             •             .                           .  .41 

Gothic  arches  described,              *             «             •             •             «  58 
Grades,         ..«•«•««.         13 

Grecian  mouldings,         ..«.«„  70 

Grounds  to  plinth,  &c.,          .             .             «             .             .  .66 

Ground  line  or  base  line,             .             «             .             ,             ,  86 

Habit  of  observation,              .              .             «             .             .  .       1 05 

Half  distance,    .             .                           .             .                           .  92 

Height,  rise  or  versed  sine  of  an  arch,            .             .             •  •         54 

Hemisphere,        .......  41 

Hexahedron  or  cube,              .             ,             -             .             «  .37 

Hexagonal  pavement  in  perspective,                                    *  94 

Hipped  roof — hipped  rafter,  .                           .             „             „  .64 

Horizontal  or  level  line,               .              .             ,             »             .  8 

"         covering  of  domes,            .             ,             .             .  .53 

Horizon  in  perspective,  ..,„,«  86 

Horseshoe  arch,         .             •             .             .             .             .  .57 

"        pointed  arch,               .             .             .             .             .  59 

Hyperbola  the  section  of  a  cone,         .                          .             .  .45 

"         To  describe  the  hyperbola  from  the  cone,         .             .  46,  48 

Hypothenuse,             .             .             .             .             .             .  .     9,  16 

"           Square  of  the                                       .             .  16 

Icosahedron,               .          '  .             .             .             .             .  .42 

Inclined  lines  in  Isometrical  drawing  require  a  different  scale,       .  81 

Incidence — -The  angle  of  incidence  equal  to  the  angle  of  reflection,  .         81 

Inclined  lines — Vanishing  point  of           ....  89 

Inclined  planes — Vanishing  point  of               .             .             .  .90 

Inscribe — To  inscribe  a  circle  in  a  triangle,    .             .             .  »         19 

"                "           an  octagon  in  a  square,        .             .  20 

"                "           an  equilateral  triangle  in  a  circle,          .  .         21 

"                "           a  square  in  a  circle,             .             .             .  21 

"                "           a  hexagon  in  a  circle,                .  21 

c<                "           an  octagon  in  a  circle,         .             .             «  21 

"                "           a  dodecagon  in  a  circle,            .            *  -         21 

Intrados  or  soffit  of  an  arch,        .             .             .             .  •           .  55 

Isometrical  drawing,              ...             •             •  .76 

"          cube,             ......  76 

"          circle,                   .             .             .             .             .  .79 

_ 


152 

PAGE. 

Isometrical  circle — To  divide  the  ....  80 

Isosceles  triangle,     ......  .9 

Joints  of  an  arch  defined,  .  .  .  55 

"     To  draw  the  joints  of  arches,  .  *  .  .  56  to  59 

Joists — Plan  of  a  floor  of  joists,  .....  65 

"      Trimmers  and  trimming         .  .  .  .  .65 

"      Tail        .......  65 

Keystone  of  an  areh,  ...  .  .         55 

Lancet  Arch — To  describe  the    .  .  .  .  .  58 

Light — Objects  to  be  seen  must  reflect  light,  .  .  .81 

"       becomes  weaker  in  a  duplicate  ratio,  &e.,  .  .  81 

"        Three  degrees  of       .....  82,  111 

Lines — Description  of     .  .  .  .  »  .  7 

Line — To  divide  a  right        .  .  .  .  .  .25 

"      To  find  the  length  of  a  curved       .  *  .  23 

"      Workmen's  method  of  doing  so^  .  .  .24 

Line  of  centres  (of  wheels,),         .....  73 

"     Pitch  line  denned,       ......         73 

"     To  draw  the  pitch  line  of  a  pinion  to  contain,  a  definite  number  ef 

teeth,  .......         74 

"     Ground  or  base  line,  .....  86 

"     Vanishing  point  of  a  line,        .  »  »  »  .89 

"     of  elevation  in  perspective,  ....  95 

Linear  perspective  defined,    ......         86 

Listely  band  or  fillet,       ....  .68 

Locomotive  cylinder,.  .  »  »  »  ».  .75 

Lozenge  defined,  .  .  »  »  .  .  10 

Major  and  minor  axes  o*  diameters,  .  .  .  .  43,  44 

Measurements  to  be  proved  from  opposite  ends,   ...  63 

Measures — Cubic     .  37 

"          Lineal  and1  superficial  ....  37 

"          of  the  surface  of  a  cube,  .  »  »  .38 

Middle  ray  or  central  visual  ray,  ....  87 

Minutes,       ........         12 

Mitre — To  fend  the  eut  of  a.  .  .  .  »  18 

Moresco  or  Saraeenie  arch,  .  .  .  .  .  .57 

Mouldings — Roman        ......  68 

"  Grecian  »*».....         70 

Obelisk  defined,  .  39 

Oblique  pyramid,       .........         39 

"      cone,      .  ' 40,45 

"      cylinder,       .»»»««.         42 
Oblong  defined,    .  »  ^  «,  »  .  10 

Octagonal  plan  and  elevation,  .  *  .  .  .67 

Octahedron,        .......  42 

Ogee  Arch  or  arch  of  contrary  flexure,  .  .  .  .57 


153 

PAGE. 

Ogee  or  cyma  reversa — Roman   .  69 

"            "          "         Grecian         ....  71 

Optical  illusion,               .                           ....  77 

Ordinate  of  an  ellipsis,           .                           ...  44 

Ovals  composed  of  arcs  of  circles,            ....  33,  34 

Ovolo — Roman          4             ...                           .  69 

"      or  Echinus — Grecian      ...                           .  TO 

Parallelogram  denned,           .....  10 

Parallel  lines,     .......  7 

"        ruler,            ......  24 

"       Application  of  the  parallel  ruler,              ...  25 
Parallelopipedon,      .......         38 

Parabola — To  find  points  in  the  curve  of  the        ...  32 

"         the  section  of  a  cone,         .             .             .             •  .45 

"         To  describe  a  parabola  from  the  cone,             .             v  46 

"         To  describe  a  parabola  by  tangents,  &c.     .             .  .49 

"         To  describe  a  parabola  by  continued  motion,  .             .  50 

u         applied  to  Gothic  arches,               .             .             .  .51 

"         Definitions  of  the  parabola,     ....  50 

Parameter  defined,    .......         50 

Pentagon  reduced  to  a  triangle,    .....  25 

"       To  construct  a  pentagon  on  a  given  line,     .  •        21 

Perimeter  the  boundary  of  polygons,                      ...  9 

Periphery  the  boundary  of  a  circle,                  .             .          :".  •         11 

Perpendicular  lines  defined,         .....  8 

"            To  bisect  a  line  by  a  perpendicular,      .             .  .14 
"            To  erect  a  perpendicular,  .             ...             .  14,  15,  16 

"            To  let  fall  a  perpendicular,        .             .             .  .  15,  16 

Perspective  view  necessary  to  shew  the  effect  of  a  design,              .  61 

"          Essay  on  perspective,      .             .                     %,-    .  .81 

"          Linear  and  aerial  perspective,             v             .             .  86 

"          plane,  or  plane  of  the  picture,       .             *             .  .         86 

"             "      must  be  perpendicular  to  the  middle  visual  ray,  87 

"          plan  of  a  square,        .  •            .             .             .             .  90,  91 

"             u   of  a  room  with  pilasters,       .             .             .  .92 

"             "  To  shorten  the  depth  of  a  perspective  plan,         .  92 

"          Tesselated  pavements  in  perspective,       .             .  .93 

"          Double  square  in  perspective,             .             .             •  94 

"          Circle  in  perspective,       .            v            .             •  .95 

u          Line  of  elevation,      .             .             .             .             .  95 

"          Pillars  with  projecting  caps  in  perspective,           .  .         96 

"          Pyramids  in  perspective,                      ...  97 

"          Arches  seen  in  front,        .             .             .             •  .98 

"               "     on  a  vanishing  plane,                .             .             •  100 
"          Application  of  the  circle,              ....       101 

"          To  find  the  perspective  plane,  &c.      .             .             .  103 

_ 


154 


Perspective  view  of  a  cube  seen  accidentally,  .  .  .104 

view  of  a  cottage  seen  accidentally,  .  .  106 

view  of  a  street,  .  .  .  .108 

Piazza  and  Cornice — Their  effect  on  the  design  for  a  cottage,  62 

Pillars  in  perspective,  .  .  .  .  .  .96 

Pitch  of  a  wheel,  ......  73 

"    circle  of  a  wheel,          ...  73 

"    line  of  a  wheel,      ......  74 

Plan — A  horizontal  section,  .  .  .  .  .60 

Plane  superficies,  ......  9 

Planes — Vanishing  ......         88 

"       Parallel  planes  vanish  to  a  common  point,  .  .  90 

"       parallel  to  the  plane  of  the  picture,  .  .  .90 

To  find  the  perspective  plane,    .  .  .  .  103 

Platband  or  straight  arch,  .  .  .  .  .55 

Platonic  figures,  ......  42 

Plinth — Section  of  parlor  plinth,       .  .  .  .  .66 

Point  of  intersection,  .  .  .  .  .  13 

"     of  contact,      .......         13 

"     Secant  point,        ......  13 

"     of  sight,         .......         87 

"     of  view  or  station  point,  ....  87 

"     Vanishing  points          .  .  .  .  .  .87 

"     Principal  vanishing  point,  ....  88 

"     of  distance,  ......         87 

Pointed  arches  in  perspective,  ....  99 

Poles  of  the  sphere,  ......         41 

Proportional  diameter  of  a  wheel,  ....  73 

circle  or  pitch  line,        .  .  .  •.  .73 

Polygons  described,         .  .  .  .  .  .  9 

"      Table  of  polygons,  .....         19 

"      Regular  and  irregular  polygons,  .  .  .         19,  20 

Polyhedrons,  .  .  .  .  .  .  37, 42 

Projecting  caps  in  perspective,  ....  96 

Protractor — Construction  of  the  protractor,    .  .  .  .30 

Application  of  the  protractor,  ...  30 

Prisms,         .......  38,  39 

Pyramid,  .......  39 

"       in  perspective,        ......         97 

Quadrant  of  a  circle,      .  .  .  .  .  .  12 

Quadrangle  defined,  .  .  .  .  .  .10 

Quadrilateral  defined,    .  .  .  .  .  .  10 

Radius — Radii,          .  . "  .  ,  .  H 

Rafters — Elevation  of  rafter,       .  ...  .  .  65 

"Hip 64 

Rampant  arch,  .  .  55 


155 

Rays  of  light  reflected  in  straight  lines,          .  .  .  83 

converged  in  the  crystalline  lens,     ...  83 

Rectangle  denned,  .  .  .  .  .  10 

Reduce — To  reduce  a  trapezium  to  a  triangle,      ...  25 

To  reduce  a  pentagon  to  a  triangle,  .  .  .25 

Reflection  of  light,          ......  81 

"        The  angle  of  reflection  equal  to  the  angle  of  incidence,       .         81 
Reflected  light  enables  us  to  see  objects  not  illuminated  by  direct  rays,        83 
Regular  triangles,  .  .  .  .  .9 

"        polyhedrons,  .  .  .  .  .37  and  42 

Requisites  for  a  country  residence,    .....         61 

Refraction  of  light,         .  .  ...  82 

Retina  of  the  eye,  ......         83 

Rhomb — Rhombus,        .  .  •  .  .  .  10 

Rhomboid,  .......         10 

Right  angled  triangle,  ....  9 

Right  line  denned,  ......  7 

"     pyramid,  ......  39 

"     cylinder,        ...  ...         40 

"     cone,  .  .  .  .  .  40,  45 

Rise  or  versed  sine  of  an  arch,          .  .  ...  .54 

Rise  or  riser  of  stairs,  .....  63 

Roof- — Hipped  .  •  .  •  .  .  .64 

"      Section  of  roof,  ......  65 

Roman  mouldings,  ......         68 

Rule  of  3,  4  and  5,  .  .  .  .  .         16,  17 

Saracenic  or  Moresco  arch,  .  .  .  .  .57 

Scale  of  chords,  ......  26 

Scales  of  equal  parts,  ......         27 

"      Simple  and  diagonal  scales,  ....         28,  29 

"      Proportional  scale  in  perspective,         .  '*  •          .  .         96 

Scalene  triangle,  .  •  •  •  .  .  9 

Scheme  or  segment  arch,       .  .  .  ,  .  .56 

Scotia  described — Roman,  ....  68 

"  "          Grecian,    ......         72 

Secant — Secant  point,  or  point  of  intersection,     .  .  13 

Seconds,  .......         12 

Sector  of  a  circle,  .  .  .  .  .  .  12 

Section — a  vertical  plan,         .  •  •  .  •  .60 

Sections  of  the  cylinder,  .....        42,  43 

Section        "         "         through  three  given  points,   .  .  .51 

"     of  the  cone,       .  .  •  .  •  .      45  to  48 

"     of  the  eye,  ......        83 

Serpentine  line,  ......  7 

Segment  of  a  circle,  .  .  .  .  .  .12 

"     To  find  the  centre  for  describing  a  segment,       .  ,  23 


156 

PAGE. 

Segment — To  find  a  right  line  equal  to  a  segment  of  a  circle,  .  .         23 

"       To  describe  a  segment  with  a  triangle,  .  .  31 

"       To  describe  a  segment  by  intersections,     .  .  .33 

"       of  a  sphere,     ......  41 

"       or  scheme  arch,     ......         56 

Semicircle,          .  .  .  .  .  •  .  12 

Semicircular  arch,    .......         56 

"         "          in  perspective,  ....  99 

Shade  and  shadow,  .....  82,  111 

Shadow  always  darker  than  the  object,     ....         87,  90 

Shadows — Essay  on  shadows,  .  .  .  .  .111 

Shading  of  circular  objects,         .  .  .  .  .  112 

Shadow — Lightest  and  darkest  parts  of  a      .  .  .  .116 

Sight— Method  of  sight,  .  .  .  .81,83 

"      Point  of  sight,  ......         87 

Sills  of  window,  .  .  .  .  .  66 

Simple  and  complex  arches,  .  .  .  .  .55 

Sine,       ........  13 

Skew-back  of  an  arch,  ......         56 

Soffit  or  intrados  of  an  arch,       .  .  .  .  .  55 

Span  of  an  arch,      .  .  .  .  .  .  .54 

Sphere — Definitions  of  the  sphere,  ....  41 

"      To  draw  the  covering  of  a  sphere,   .  .  .  .41 

Springing  line  of  an  arch,  .....  54 

Square,         .  .  .  .  .  .  .  .10 

Square  corner  in  a  semicircle,      .  .  .  .  .  15 

"          "     by  scale  of  equal  parts,  .  .  .  .16 

"     of  a  number,        ......  16 

"     of  the  hypothenuse,    .  .  .  .  .  .16 

Stairs — To  proportion  the  number  of  steps  of  stairs>        .  .  63 

Station  point  or  point  of  view,  .  .  .  .  .87 

Stiles  of  sash,  &c.  ......  66 

Step  or  tread,  .......         63 

Straight  or  right  line,     ......  7 

"       arch  or  plat  band,    ......         55 

Street  in  perspective,       .  .  .  .  .  .  108 

Subtense  or  chord,     .  .  .  .  .  .  .12 

Summit  of  an  angle,        .....  ^.  8 

"      of  a  pyramid,  ......         39 

"      of  a  cone,  .....  40, 44 

Superficies  or  surface,  .....  9 

Supplement  of  an  angle  or  arc,    .  .  .  .  .  13 

Table  of  the  names  of  polygons,        .....         19 

"     "  the  angles  of  polygons,  ....   Plate    10 

Tail  joist,  .......         65 

Talon  or  Ogee — Roman,  .....  69 


157 

PAGE. 

Talon  or  Ogee — Grecian,      .  .  .  •  .  .71 

Tangent  defined,  .  .  .  .  •  •  13 

Teeth  of  wheels — To  draw  the  teeth  of  wheels,  .  .  .73 

"          "  Pitch  of  the  "  ...  73 

"          "  Depth  of  the  "  ....         73 

Tesselated  pavements  in  perspective,        ....  93 

Tetragon  defined,      .  .  .  .  •  •  .10 

Tetrahedron  one  of  the  regular  solids,      ....  41 

Torus  described,      .  ......         68 

Trapezium  defined,         ......  10 

"          reduced  to  a  triangle,     .  .  .  .  .25 

Trapezoid  defined,  ......  10 

Transverse  axis  or  diameter,  .  .  •  .  •  42, 43 

Tread  or  step,  ......  63 

Trigons  or  triangles,  ...... 

Trisect — To  trisect  a  right  angle,  •  •  •  •  18 

Trimmers  and  trimming  joists,  .  ...»         65 

Truncated  pyramid,         ...... 

"         cone,         .  .  .  •  •  .45 

Tudor  or  four  centred  arch,         .....  59 

Vanishing  points,      ..••••• 
"         Principal  vanishing  point,  •  •  • 

"         planes,     . 
Versed  sine  of  an  arc,     .  .  •  •  • 

"       "    or  rise  of  an  arch,  ...  .54 

Vertex  of  a  triangle,        .  .  •  •  • 

"     of  a  pyramid,  ...  . 

"     of  a  cone,  ..••••  44 

"     of  a  diameter  of  the  ellipsis,  •  • 

"     Principal  vertex  of  a  parabola,      .  .  .  50 

"     of  a  diameter  of  the  parabola,  ....         50 

Vertical  or  plumb  line,    ..•••• 

"      coverings  of  domes, .  .  ... 

Visual  angle,  .  •  •  •  •  • 

"      rays,  ....-•• 

Voussoirs  of  an  arch,       .  ... 

Wheel  and  pinion — Drawing  of  a 

«  "          To  proportion  the  teeth  of  a  .  •  ?4 

Wheel  viewed  in   perspective, 
Windows — Details  of  windows  .  •  .  .        65,  66 


159 


INDEX 

TO    THE    ESSAY    ON    OOLOE. 


PAGE. 

Absorption  of  light  by  all  bodies,       •            •            •            •  .        •     127 

Advancing  colors,             ...•••  125,  134 

Application  of  colors  to  Geometrical  Drawings,         .            . ,  .              139 

Artificial  light,  its  effect  on  colors,          ....  126,  135 

Ash-color,  slate,  lead-color,  etc.,  modifications  of  gray,         .•  .              133 

Auburn,  hazel  and  dun,  modifications  of  brown,            .             .  .     133 

Black,  the  total  absence  of  light,                   .            .             .  .               123 

"      How  the  term  is  usually  applied,            ...  .      123 

"      Neutral — Compounded  of  the  three  primaries,         .  .              132 

Blue,  its  properties  and  effects  on  other  colors,               .  .      130 

"      Light  tints  of — known  as  pearl  white,  silver  gray,  &c.,  •              133 

Bodies,  have  no  inherent  color,       .....  .      127 

Brass  or  Bronze,  the  colors  for  indicating  them,      .            .  .                142 

Brick,  the  colors  used  to  represent  it,    .             .             .            .  .      141 

Broken  colors,  the  general  acceptation  of  the  term,              .  .132,  134 

Browns,  Composition  and  qualities  of    .            .            .            .  .      133 

Citron  or  yellow  hue,  ......        124,  131 

Color,  the  term  defined,               .            .            .            ...  .118 

Colors,  Broken  colors,           ......  132 

"       Contrasting  colors,           .....  .      120 

"       Complementary  colors,          .             .            .            .    .  .                120 

"       Cool  colors,  retreating  from  red  toward  black,    .            .  125,  134 

"       Conventional  colors,  to  denote  the  construction,       .  .                141 

"       How  to  use  them,             .....  142,  143 

11       Homogeneous — yellow,  red  and  blue,           .           <•   :  .                122 

11      List  of — for  architectural  and  mechanical  drawings,       .  .      139 

11       Neutralizing  powers  of  the  primary              .            .  .                124 

"       Nomenclature  of  colors  uncertain,           .            .            .  133,  134 

11       Neutral  and  semi-neutral     ....  132,  133,  134 

"       Primary 123,  134 

11       Secondary  and  tertiary         .                         .            .  .        124,  134 

"       Of  the  solar  spectrum,     .            .            .            .            •  .      121 

"       Scales  of 123,  124,  125 

"       Transparent  and  opaque              ....  •      139 

Copper,  the  color  used  for  indicating  it, 

Crimson,  pink,  peach-blossom,  rose-color,  lilac,  violet,  &c.,  combinations  of 

red  and  blue,         ......  133 

Decorations,  that  are  to  be  viewed  by  artificial  light,  require  peculiar 

treatment,        ......  •      126 

Definitions' of  terms,  recapitulated,              ....  134 


160  INDEX   TO    ESSAY    ON    COLOR. 

PAGE. 

Detail  Drawings,  how  to  make  and  tint  them,  .             .             .  136 

Drawing  Boards,  the  best  way  to  make  them,         .            .            .  136 

"              paneled  for  stretching  the  paper  on,                .  .      137 

Drawing  Instruments  and  requisites  for  geometrical  drawing,         .  138 

"                  methods  of  using  them,    ...  .      138 

Drawing  Paper,  Regular  sizes  of                  .             .            .            .  137 

"             which  side  of  the  sheet  to  draw  on,     .            .  .137 

Method  of  stretching         ....  137 

"             to  back  it  with  muslin,              ...  .      145 

"             to  correct  a  greasy  or  glazed  surface,         .            .  143 

Drawing  Pins,  for  holding  the  paper  on  the  board,       .             .  .      136 

Engravings,  to  clean  and  mount  them,         ....  145 

Experiments  on  light  by  Sir  Isaac  Newton,        ...  .      121 

"                "        by  Sir  David  Brewster,       •  .            .             .  122 

"           with  red  and  green  wafers,             .            .            .  .120 

"           with  a  circular  disk,  (note,)     ....  124 

Flowers  which  indicate  the  primary  colors,        .            .            .  .117 

"       Blue  natural  flowers  extremely  rare,          .            .            .  117 

Geranium,  the  red  geranium  to  represent  the  primary  red,       .  .117 

Geometrical  Drawings  described,     .....  136 

Glazing,  the  laying  a  transparent  color  over  another,                .  .     139 

Gold-color,  giraffe,  scarlet,  &o.,  combinations  of  yellow  and  red,     .  133 

Gray,  Neutral     .......  123,  134 

Green,  its  properties  and  effects,      .            .             .             .            .  131 

Grass-green,  Emerald-green,  etc.,            ....  .      133 

Harmony  of  colors,  definition  of  the  term,  .  .  .119,  120 

Hue,  definition  of  the  term,       .            .  -          .            .  .         .  .119 

Illustration  by  drops  of  color,          .             .            .            .            .  118 

Ink,  the  common  writing  ink  should  not  be  used  for  drawing  purposes,  .      138 

11     India  or  China  ink,      ......  138 

"     Carmine  aids  it  to  flow  more  freely,           .            .            .  .138 

Iron,  wrought  and  cast,  the  colors  for  indicating,   .            .            .  141,  142 

Jasmine,  the  yellow  to  represent  the  primary  yellow,                .  .117 

Keeping,  definition  of  the  term,        .             .             .             .             .  119 

Key  or  Tone,  the  prevailing  hue  of  a  landscape,  etc.,    .            .  .119 
Lakes,  Lilac,  see  crimson.     ...... 

Lead  and  Tin,  the  colors  for  indicating  them,                .            ."  -  .     142 

Leather,                    "                     "                .             ...  142 

Light,  shade  and  shadow,           .            .            .            .  -          .  .120 

"      and  shade,  represented  by  white  and  black,              .            .  123 

"      Artificial — its  effects  on  colored  surfaces,             .            .  .      126 

"            "         always  inclines  to  yellow,               .             .             .  126 

"      Absorption  and  reflection  of        ....  .      127 

Lights,  in  water  color  drawings,  should  be  represented  by  the  surface  of 

the  paper,            ......  .      140 

Manipulation  of  the  colors,               .....  148 

Materials  should  be  of  the  best  quality,             ...  .      146 

Melody,  the  term  defined,     ......  120 


INDEX    TO    ESSAY   ON    COLOR.  161 

PAGE. 

Melody  and  Harmony,  applied  to  colors  as  in  music,     .                .  .      120 

Melodizing  colors,  those  which  follow  a  certain  natural  scale,           .  120 

Mounting  drawings,         ....  145 

Murrey,  puce,  etc.,  modifications  of  maroon,            .             ,             .  1^3 
Nature,  All  the  various  tints  and  hues  of — produced  by  three  colors,  .      117 

Natural  scale  of  colors,     ......  125 

Neutral  blacks  and  grays,  semi-neutral,  etc.,  .  .        132,  133,  134 

Neutral  grays,  the  gradations  between  white  and  black,             .  .123 

Neutral  tint,  a  mixed  pigment  for  shading,  etc.,        .            .            .  142 

Obstacles  to  be  removed  from  the  path  of  the  learner  at  once,  .      145 

Olive  or  blue  hue,       .......  124,  132 

Opacity  and  transparency,  not  absolute  terms,               .            .  .127 

Orange,  its  properties  and  effects,      .            .            .            .            .  129 

Perspective  views  if  fancifully  colored,  often  convey  a  wrong  impression 

of  the  building  to  be  erected,            .            .            .  141 

Pencils,  The  qualities  required  in  good  drawing             .             .  .      138 

should  always  be  cut  with  a  keen  knife,                   .            .  138 

Pigments  often  act  chemically  on  each  other,                .               ,  .      139 

by  different  makers  seldom  correspond  in  hue,    . .           .  134 

the  fewer  used  to  make  a  tint  the  better,       .            .  .      140 

Pink,  composed  of  red  and  blue,     .....  133 

Plans  and  sections  should  be  colored  to  show  the  materials  designed  for 

the  construction,           •             •             .            .             .  .      140 

Positive  colors,            .......  134 

Primary  colors,  homogeneous,  etc.,          ....     122,  124,  134 

Prism  of  glass,  for  the  refraction  of  light,                .             .             .  121 

Purple,  its  properties  described,              ....  .      130 

Quality  of  materials,             .             .             .             .             .             .  146 

Refraction  of  light  in  the  solar  spectrum,           .            .            .  .121 

Eefrangibility  of  the  primary  colors  unequal,           .             .               .  121 

Reflections  Effects  of — from  colored  surfaces,     .             .             .  .      12'6 

Red,  its  properties  and  effects,         .  129 

Red  Light  tints  of — known  as  rose  white,  carnation,  coquelicot,  etc.,  .      133 
Retreating  colors,                    .             .             .             .             .             .125,  134 

Rose  color,  see  crimson 

Rubble  stone  walls,  how  to  color  them  in  sections,           .            .  .      141 

Russet  or  red  hue,                 *             .  • '  •         .             .             .             .  124,  132 

Sage  The  blue — to  represent  the  primary  blue,               .            .  .117 

Scarlet,                       129,  133 

Scales  of  the  colors,        ...             ...      123,  124,  125 

Secondary  colors,       ....'...  124,  134 

Sections  and  plans,  colored  to  show  the  construction,      .             .  .      140 

Shade,  definition  of  the  term,           ....  119 

Shading  detail  drawings  by  ruling  parallel  lines,            .  .136 

Shadows  on  plans  and  sections  to  be  at  an  angle  of  45°,     .             «  140 

Solar  spectrum,    .             .             .             .             .             .             .  .121 

Steel  or  iron,  the  colors  for  indicating  them,             .             .             .  142 

Stone,              "                "                "                   ...  .      141 

21 


162 


INDEX    TO    ESSAY    ON    COLOK. 


Tertiary  colors,          ....... 

Theory  of  color,  ..•••• 

Tint,  the  term  defined,         .            .            •            .             .  . 

Tone,      "  " 

Tracing  cloth,  for  copying  drawings,  .  •  •  • 

Translucent  substances  transmit  light  but  not  vision,    .  , 

Transparent  media,  permit  light  and  vision  to  pass  freely,  • 

Transparency  and  opacity,  only  relative  terms,  .  . 

Violet  in  the  solar  spectrum,  .  .  •  . 

"      composed  of  red  and  blue,  .  .  . 

Vulcanized  Rubber,  colors  to  represent  it,  .  •  • 

Wafers,  experiment  with  red  and  green  wafers,  .  . 

Warm  colors,  red,  orange,  yellow,    ..... 
Wash,  to  correct  the  greasiness  of  paper  to  be  colored,  . 

White  the  representative  of  light,  its  properties  and  effects,  . 

White  light  may  be  resolved  into  the  three  primary  colors, 
Woods,  how  to  color  them  in  sectional  drawings,     .  .  . 

Yellow,  its  properties  and  the  effect  it  produces,  . 

"       its  character  entirely  changed  on  coarse  fabrics,     .  . 

"       should  not  be  used  in  large  masses,        .  .  . 

"      Light  tints  of — called  primrose,  straw  color,  cream  color,  etc., 


PAGE. 

124,  134 
.     117 

119 
.  119 

146 
.  127 

127 
.  127 

121 
.  133 

142 
.  120 

125,  134 
.     143 

126 
.  123 

142 
.  128 

128 
.  128 

133 


DIRECTIONS    TO    THE    BINDER. 


Plate  54  to  face  the  title  page. 

"    1,2,3  to  follow  page 

tc 


"  4,5 
"  6,  7 
"  8,9 

11     10,  11 

"  :  1-2,  13 
"     14,  15 


16,  17,  18,  19  " 

20,  21  " 

22,  23  " 

24,  25  " 

26,  27,  28,  29  " 

11    30,  31,  32,  33  " 


8 
16 

20 
24 

32 
36 
40 
44 
48 
52 
56 
64 


Plate  34,  35,  36,  37  to  follow  page 
"    38,  39 
"    40,  41 


"     42,  43 
"    44,  45 


" 


46,47 
48,  49 
50 


to  face 


"  51 

"  52 

"  53 

"  55 

"  56 


7,2 

76 

80 

84 

92 

96 

100 

102 

103 

105 

106 

113 

116 


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